btSl"      ■       •••.-•  :=— ^  ••-,*'. .V,.-.    ' 


I%^\ 


IN   MEMORIAM 
FLORIAN  CAJORI 


♦  • 


AN 


ELEMENTARY   TREATISE 


ON 


CURVES,   FUNCTIONS,   AND  FORCES. 


VOLUME    SECOND; 


CONTAINING 


CALCULUS    OF    IMAGINARY   QUANTITIES,   RESIDUAL 
CALCULUS,  AND  INTEGRAL  CALCULUS. 


By  benjamin  PEIRCE,  A.  M. 

Perkins  Professor  of  Astronomy  and  Mathematics  in  Harvard  University. 


BOSTON: 

JAMES    MUNROE    AND    COMPAJMY 

1846. 


Entered  according  to  Act  of  Congress,  in  the  year  1846,  by 

James  Munroe  and  Company, 

in  tho  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts 


BOSTON: 

PRINTED  BY  THURSTON,  TORRY  &  CO. 
31  Devonshire  Street. 


CONTENTS. 


BOOK   III. 

CALCULUS  OF  IMAGINARY  QUANTITIES. 
CHAPTER  T. 

MODULUS  AND  ARGUMENT         ...... 

CHAPTER  n. 

IMAGINARY  INFINITESIMALS 

CHAPTER  m. 

IMAGINARY  ROOTS  OF  EQUATIONS     ..... 

CHAPTER  IV. 

IMAGINARY  EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS  22 

CHAPTER  V. 

IMAGINARY  CIRCULAR  FUNCTIONS    .....  25 

CHAPTER  VI. 

REAL  ROOTS  OF  NUMERICAL  EQUATIONS  ....  31 


BOOK     IV. 

RESIDUAL  CALCULUS. 
CHAPTER  L 


RESIDUATION 


iV!:?.Of>5^:57 


3 
12 
13 


43 


IV  CONTENTS. 

CHAPTER  II. 

DEVELOPMENT  OF  FUNCTIONS,  WHICH  HAVE  INFINITE  VALUES  52 


BOOK    V. 

INTEGRAL  CALCULUS. 
CHAPTER  I. 

INTEGRATION  ........  63 

CHAPTER  II. 

INTEGRATION  OF  RATIONAL  FUNCTIONS  .  .      •      .  69 

CHAPTER  III. 

INTEGRATION  OF  IRRATIONAL  FUNCTIONS         ...  77 

CHAPTER  IV. 

INTEGRATION  OF  LOGARITHMIC  FUNCTIONS    .     .     .   101 

CHAPTER  V. 

INTEGRATION  OF  CIRCULAR  FUNCTIONS     .  .  .  .116 

CHAPTER  VI. 

RECTIFICATION  OF  CURVES       ......       124 

CHAPTER  VII. 

QUADRATURE  OF  SURFACES 166 

CHAPTER  Vm. 

THE  CURVATURE  OF  SURFACES 185 

CHAPTER  IX. 

THE  CUBATURE    OF  SOLIDS        ......       197 

CHAPTER  X. 

INTEGRATION  OF  LINEAR  DIFFERENTIAL  EQUATIONS  .       207 


CONTENTS.  T 

CHAPTER  XL 

INTEGRATION  OF  DIFFERENTIAL  EQUATIONS  OF  THE  FIRST 

ORDER        .........       239 

CHAPTER  Xn. 

INTEGRATION  OF  DIFFERENTIAL  EQUATIONS    OF    THE    SEC- 
OND ORDER  ........       274 

CHAPTER  Xni. 

PARTICULAR  SOLUTIONS  OF  DIFFERENTIAL  EQUATIONS        .       287 


CORRECTIONS, 


Page 

line 

25 

12 

26 

it 

t( 

(107) 
(110) 
(111) 
(114) 

29 

(138) 
(139) 
(140) 

32 

(149) 

<( 

(151) 

48 

7 

53 

23 

54 

14 

55 

(217) 

64 

(247) 

65 

(252) 

66 

2      I 

89 

(373) 

90 

(380) 

92 

1 

t< 

2 

95 

14 

106 

(439) 
(443) 

107 

(19  and  20) 

) 


for  read 

n  7C  (n-\-fi)  71. 


2B  B. 

f.(a-\-i)  dcf.(a  +  i) 


t  I 

f(a-^i)  dcf.(a  —  i) 


t  I 

b^—2ab  —  a2  b^—3ab. 

finite  infinite. 
31  33. 

(^  +  a)  f.(x  +  a). 


/•  ^0  f'  ^i  — /•  ^0  • 

3 

4 

m  +  l-\-np  —  ns  m+l+n/j  —  ns 


The  fraction  is  to  be  multiplied  by  (1 z^  +  —  z^.) 

"'  4  a  4a2      ' 


s  n  s 

pP—i  bP—^. 

bx^  b  x^ 


a  a 

10  ^10 

biomial  binomial. 

I            2A  +  1  2A-1. 

108        (458)     The  first  and  second  members  are  to  be  divided  by  ^. 

113  (501)                                71  +  1  -    n  — 1. 

114  (504)                                n  +  1  n  — 1. 


Vlll  CORRECTIONS. 

Pagt           line  for  read 

119          (543)  ——  +    —  . 

121            5  1  i 

125  9  e2z^g-2,  e2*_2^e-2*. 

126  18  or  F'  T  or  F'  T. 

127  —1  P'  p, 

"  11       Add^  or  are  supplements  of  each  other  (fig.  2). 

128  2  and  3  pi  p. 

129  (574)  e  e". 
134        (612)  t  V. 

(618)  ±  d=^. 

136  11  are  is. 

137  J  3  (635)  (630). 

138  (643)  fl"  and  y'/  ^^  and  i^^ . 
"         (648)  f-  fl 

139  (649)  If"  if^. 
"         (652)  f-  jfl 

140  (658)  (618)  (658). 
"  9  fig.  5  fig.  6. 
"19  X  V- 

143         (680)  2e  4  e. 

"      13  and  15  F'  F". 

148      2  /;p  /;. 

149           2  tangent  perpendicular. 

m/«Qo\  <P  sin.  /9  cot.  a 

(^^^)  I  gVsin.^cot.a. 

"          (735)  (p  (p  cot.  a. 

152  (738)  &  (739)  (p  ip  cot.  a. 

154  (751)  X'o  Xq'. 

155  24  time  line. 
164         (789)  1  +  e2  COS.  2  a  cot.^  «. 

"          (792)  l  +  e^cos.  2  a  cot.^ «. 

170            3  angular  annular. 

274         (24  e)  Dtrp=q  +  lDtX          Dtp  =  Q-\-iDt  x. 


B  O  O  K    1 1 1 . 
CALCULUS    OF   IMAGINARY  aUANTITIES. 


BOOK    III. 

CALCULUS    OF    IMAGINARY    QUANTITIES. 


CHAPTER   I. 


MODULUS    AND    ARGUMENT. 


1.  The  general  form^  to  which  geometers  attempt 
to  reduce  all  imaginary  expressions,  is  that  of  a  bino- 
mial, in  lohich  one  term  is  real  and  the  other  term'  is 
the  product  of  a  real  factor  by  the  imagiiiary  factor 

v-i. 

a.  Thus,  if  A  denote  the  real  term,  and  B  the  real  factor  of 
the  imaginary  term,  this  binomial  type  of  imaginary  quanti- 
ties is 

A  +  BV-l-  (1) 

b.  As  this  expression  is  imaginary,  all  operations,  such  as 
addition,  multiplication,  &c.,  performed  upon  it  or  by  it,  are 
wholly  devoid  of  their  usual  meaning,  and  may  admit  of  any 
conventional  interpretation.  But,  then,  rules  must  be  adopted 
for  performing  the  operations  which  shall  be  consistent  with 
this  interpretation ;  or,  reciprocally,  the  rules  for  performing 
the  operations  may  be  assumed  at  pleasure,  provided  that  a 


IMAGINARY    QUANTITIES.  [b.  III.  CH.  I. 


Values  of  modulus  and  arffument. 


mode  of  interpreting  the  operations  and  the  results  is  adopted, 
which  is  consistent  witli  the  rules.     Now,  if  we  take 

m^  =  —  1,     that  is,     m  ^i  s/ —  1 ;  (2) 

(1)  becomes  A  -\-  B  ?n;  (3) 

and  all  algebraical  operations  may  be  performed  according  to 
the  usual  rules  upon  (3),  without  any  regard  to  the  imaginary 
value  of  m,  provided  that  the  results  are  interpreted  consist- 
ently with  this  imaginary  value  of  w,  and  the  real  value  of  m^, 
which  is  —  1. 

2.  The  modulus  of  an  imaginary  expression  of  the 
form  (1)  is  the  positive  square  root  of  the  sum  of  the 
squares  of  its  real  term^  and  of  the  real  factor  of  its 
imaginary  term-. 

The  argument  of  this  imaginary  expression  is  the 
angle,  whose  tangent  is  equal  to  the  quotient  of  the 
real  factor  of  its  itnaginary  term  divided  by  the  real 
terin. 

Thus,  if  R  denotes  the  modulus  of  (1),  and  ^  its  argument, 
we  have 


R  =  S/{A2  +  B^),  (4) 

B 
~A' 


tang.  ^  =  — .  (5) 


3.  Corollary.  If  A  -and  B  are  represented  by  the  sides  of 
a  right  triangle,  R  is  the  hypothenuse,  and  ^  is  the  angle  op- 
posite to  B^     Hence,  by  trigonometry, 

A=^  R  cos.  &,  (6) 

B=zR  sin.  ^  ;  (7) 


«§>   6.]  MODULUS    AND    ARGUMENT. 

Argument  of  any  real  quantity. 


and  the  value  of  (1)  becomes 

R  (cos.  6  +  sin.  ^.  v'  —  1).  (8) 

4.  Corollary.  Since  two  angles,  which  differ  by  two  right 
angles,  have  the  same  tangent,  there  are  two  values  of  fl  less 
than  four  right  angles,  which  satisfy  (5)  ;  and  of  these  two 
values,  that  one  is  to  be  selected  which  agrees,  in  the  signs 
of  its  sine  and  cosine,  with  (6)  and  (7).  Any  angle,  which 
differs  from  the  value  of  &  thus  found  by  four  right  angles,  or 
by  any  multiple  of  four  right  angles,  may  also  be  taken  as  a 
value  of  <5.  Thus,  if  q^  is  this  least  positive  value  of  6,  the 
general  value  of  6  is 

5  =  ^0  ±  2  n.^,  (9) 

in  which  n  is  any  integer,  and  re  is  the  ratio  of  the  circumfer- 
ence to  the  diameter. 

5.  Corollary.  When  the  imaginary  part  of  (1)  vanishes,  we 
have 

.B  :=  0,     sin.  ^  =  0;  (10) 

so  that      ^Q  =:  0,     a=:d=2w7r,  COS.  ^  =  1 ;  (11) 

or  &o^n,      3  =  zfc  (2  W-(-  1)  TT,   COS.  5  =:  —  1.      (12) 

and  (11)  corresponds  to    the   case  of  a  real  positive 
quantity  J  (12)  to  that  of  a  real  negative  quantity. 

6.  Corollary.  When  the  real  term  of  (1)  vanishes, 
we  have 

^  =  0,    COS.  6  =  0,    ^0  =  J  TT  or  =  J  TT,  (13) 

whence 

fl  =  zhj^±2n^  =  ±(2  7i±})^,  sin,5  =  ±1.    (14) 


IMAGINARY    Q,UANT1T1ES.  [b.  III.  CH.  I. 

Equal  imaginary  quantities. 


7.  Theorem.  When  the  quantity  represented  by  (1) 
vanishes,  the  real  and  the  imaginary  part,  and  the 
moduhis,  are  each  equal  to  zero,  while  the  argument  is 
indeterminate. 

Proof.     For  if 

^4-jBV  — 1  =  0,  (15) 

we  have  ^  =  —  JB  \/ —  1  ;  (16) 

that  is,   a  real  quantity  equal  to   an  imaginary  one,   which  is 
impossible,  and  (16)  cannot  be  satisfied,  unless  we  have 

^  =  0,     jB^O;  (17) 

whence,  by  (4),  R  —  Q,  (18) 

and,  by  (5),  &  is  indeterminate. 

8.  Theorem.  When  hvo  imaginary  quantities  are 
equals  their  real  and  imaginary  parts  are  separately 
equal,  and  they  have  the  same  moduhis  and  argument. 

Proof.     For  the  equation 

A+B^— l  =  A'  +  BW—  ^,  (19) 

gives,  by  transposition, 

A  —  A'  -[-  {B  —  B')  s^  —  \  =  ^.  (20) 

Hence,  by  the  preceding  theorem,  « 

A  —  A'  =  (),     B  —  B'=:Oy 
or,  A=:A',  B  =  B;  (21) 

whence,  by  (4  and  5), 

R  =  R',  &  =  &'.  (22) 


<^    11.]  MODULUS    AND    ARGUMENT.  7 

Conjugate  quantities.  Imaginary  product. 

9.  Two  imaginary  quantities  are  conjugate  to  each 
other,  when  they  have  the  same  modulus,  and  when 
their  arguments  only  differ  in  being  of  contrary  signs. 

Thus  the  conjugate  of  (8)  is 

R  [cos.  (—  6)  4-  sin.  (—  a) .  ^  _  1]  ;  (23) 

or,  by  trigonometry, 

i?  (cos.  ^  —  sin.  5.v/— 1).  (24) 

10.  Corollary.  Two  imaginary  quantities,  which  are 
conjugate  to  each  other,  differ  only  in  the  sign  which 
precedes  the  imaginary  part. 

Thus  A  4-  Bs^  —  1  and  A  —Bs/—  1  are,  by  (8  and 
24),  conjugate  to  each  other. 

11.  Theorem.  The  modulus  of  the  product  of  sev- 
eral imaginary  quantities  is  equal  to  the  product  of  the 
moduli  of  the  factors,  and  the  argument  of  the  product 
is  equal  to  the  sum  of  the  arguments  of  the  factors. 

Proof,     a.   When  there  are  two  factors 
R  (cos.  &  +  sin.  a.  V —  1)  and  7J'(cos.  6'4-sin.5'.\/ — 1),   (25) 
the  product  is 

R  R'  [cos.  6  cos.  &'  —  sin.  q  sin.  S'] 
+  (sin.  &  COS.  6'  4"  ^^^-  ^'  COS.  &)  \/  —  1,  (26) 

which,  by  (26  and  28  of  Trig,),  becomes 

R  R  [cos.  (^  +  tv)  -f  sin.  {d  +  6') .  V—  1] ;         (27) 

so  that  its  modulus  is  the  product  of  the  two  moduli,   and  its 
argument  is  the  sum  of  the  two  arguments. 


8  IMAGINARY    QUANTITIES.  [b.  III.   CH.  I. 

Imaginary  power. 

h.  A  third  factor  miglit  be  multiplied  by  (27)  in  the  same 
way,  that  is,  by  multiplying  its  modulus  by  the  new  modulus, 
and  adding  to  its  argument  the  new  argument ;  and  this  pro- 
cess might  be  extended  to  any  number  of  factors. 

12.  Corollary.  If  the  factors  are  all  equal,  the  pro- 
duct becomes  a  power  ;  whence  the  modulus  of  a  posi- 
tive integral  power  of  an  imaginary  quantity  is  the 
same  power  of  its  modulus,  and  the  argument  of  the 
power  is  the  product  of  its  argument  hy  the  exponent  of 
the  power. 

Thus 
[/2(cos.6-|-sin.aV— l)]'^=-K"(cos.W5-fsin.w5.V— 1)-(28) 

13.  Corollary.  When  jR  =  1,  (29) 
(28)  becomes 

(cos.  6  +  sin.  a.  s/ —  1)"  i:r  (cos.  n  &-\-  sin.  n  6.  y' —  1).  (30) 

Reversing  the  sign  of  ^ 
(cos.a  —  sin.a.V — l)'^  =  (cos.  n  a  —  sm.n^.^/ —  1).  (31) 

14.  Corollary,  Half  the  sum  of  (30  and  31)  is 

cos.n  a= J(cos.a-|-sin.a V— -1 )"  +  J(cos.  5  — sin.a.  /y/— 1  )n.  (32) 

Half  the  difference  of  (30  and  31)  is 

sin.  w  a.  y' —  1  =  J  (cos.  a.  -|-  sin.  ^.  \f  —  1)" 

—  J  (cos.  d  —sin.  5.  V  —  1)"        (33) 


<§)  18.]  MODULUS    AND    ARGUMENT.  9 

Product  of  two  conjugate  factors.         Imaginary  quotient. 

15.  Corollary.    By  development,  (32)  becomes 

nln  —  1 )  _      .     ^ 

cos.w<5  =  COS."  (3 :j — ^ — -  COS.™  — ~(5  sin. 2  d 

,   7i(n  — l)(w  — 2)(w  — 3)  .      .    ^ 

_[_  _i_^ — 11 TT^-i cos."-4  &  sin.*  d  —  &c.    (34) 

By  developing  and  dividing  by  \/  —  1,  (33)  becomes 

sin.  w  ^  zz:  71  COS."— ^  ^  sin.  ^ 
n(7^_l)  (n  — 2) 


1  .     2 .     3 


cos."-3(3  sin.3(3 -}-&c.        (35) 


16.  Corollary.  The  reverse  of  ^^^  12  is,  that  the 
modulus  of  a  positive  integral  root  of  an  imaginary 
quantity  is  the  same  root  of  its  modulus,  and  the  argu- 
ment of  the  root  is  the  quotient  of  its  argument  divided 
by  the  exponent  of  the  root ;  that  is,  since  roots  are 
fractional  powers,  the  rule  of  §  12  extends  to  the  case  of 
positive  fractional  powers. 

17.  Corollary.  The  product  of  two  conjugate  factors 
is  equal  to  the  square  of  the  modulus. 

For,  in  this  case,  (23  and  27)  give 

6  +  ^'  =  ^  — a=zO,     RR'=R2.  (36) 

18.  Corollary.  The  reverse  of  ^  11  is,  that  the  mo- 
dulus of  a  quotient  is  equal  to  the  quotient  of  the  Tnodu- 
lus  of  the  dividend  divided  by  that  of  the  divisor ^  and 
the  argument  of  the  quotient  is  equal  to  the  argument 
of  the  dividend  diminished  by  that  of  the  divisor. 


10  IMAGINARY    QUANTITIES.  [b.  III.   CH.  I. 

Imaginary  power. 

Thus 

R'{cosJ'-}-smJ'.\/—\) 
RicosJ-{-s'in.6.\/—l) 

=  ^'[cos.(^'— ^)  +  sin.(5'  — ^)V— 1].  (37) 


19.  Corollauj.     When  ^'  =  0,  and  R'  —  \  \  (38) 

(37)  becomes 

z=-i  [cos.(-^)  +  sin.(-a)V-l], 


jR(cosJ  +  sin.5V— 1        ^ 
or 

[i2(cos.^+sin.^.V— l)]-^=-R"^[cos.(— ^)+sm.(— 5)V— 1] 

—  i2-i(cos.5  — sin.^V— 1);  (39) 
and  raising  to  the  wth  power,  by  means  of  (30), 

[jR(cos.^4-sin.^V— l)]~"=^~"[cos.(-n^)+sin.(-w5)V— 1] 

=  K-™(cos.w^— sin.w^^/ — 1);  (40) 

that  is,  the  rule  of  <§>  12  may  he  extended  to  the  case  of 
negative  powers. 

20.  Corollary.    The  rule  of§  12  may,  then,  be  ex- 
tended, by  §  1,  to  all  powers,  real  or  imaginary. 

21.  Problem.    To  find  the  modulus  and  argumerrt  of 
the  sum  or  difference  of  several  imaginary  quantities. 

Solution.    Let  the  given  sum  or  difference  be 

r(cos.^4-sin.^V— l)i:r'(cos.^'+sin.^'.\/— 1)±  &c.,    (41) 

and  let  R  be  its  modulus,   and  ©  its  argument;  we  have  by 
(4  and  5)  and  by  (9  and  29  of  Trig.) 


<5>  23.]  MODULUS  AND  ARGUMENT.  11 

Imaginary  sura  or  difterence. 

122=  (r  COS.  ^±r'cos.  a'i  &c.)2-j- (r  sin.  ^  ±  r' sin.  6'zh  &c.)2 

_  ^2  _|.  r'2  +  &,c.  zb  2 rr' COS.  (^  —  ^')  ±  &c.,  (42) 

r  sin.  6  rt  r'  sin.  d'  r±z  ^c.  .  ^  ^ 

tan.  0  =: — ; T— - — J —  (43) 

r  cos.  6  i  r  cos.  5'  i  &/C.  ' 

22.  Corollary.    Since  every  cosine  is  less  than  unity,  (42) 
gives  R^  <  r2  -|-  r'^  -|-  &,c.     -\-2rr'-\-  &c., 

or  i22  <  (;.  _|_  ,./  ^  &c.)2, 

*or  JR    <   r  +r'  +  &,c.;  (44) 

that  is,  the  modulus  of  the  sum  or  difference  of  several 
imaginary  quantities  is  less  than  the  sum  of  their 
moduli. 

23.  Corollary.    When  there  are  only  two  terms  in  (41), 
(42)  becomes 

2J2  -_  ^2  _j.  y./2  _t-  2  r  r'  cos.  (a  —  ^') ;  (45) 

and,  therefore,  R^  >  r^  -{- r'^  —2r  r', 

or  R    >r  —  r';  (46) 

that  is,  the  modulus  of  the  sum  or  difference  of  two 
imaginary  quantities  is  greater  than  the  difference  of 
their  moduli. 


12  IMAGINARY    QUANTITIES.  [b.  III.  CH.  II. 

Imaginary  infinitesimal. 


CHAPTER   II. 


IMAGINARY    INFINITESIMALS. 


24.  An  imaginary  infinitesimal  is  an  imaginary  quan- 
tity, whose  modulus  is  an  infinitesimal. 

The  order  of  an  im^aginary  infinitesimal  is  the  same 
with  that  of  its  modulus. 

25.  Corollary.  It  follows  from  Chapter  II.  of  the 
Differential  Calculus,  and  the  preceding  Chapter,  that 
all  the  propositions,  lohich  have  hitherto  been  investi- 
gated respecting  real  infinitesimals^  may  he  extended  to 
imaginary  infinitesimals. 


«§>  27.]  ROOTS    OF    E(iUATIONS.  13 

Roots  of  a  binomial  equation. 


(48) 


CHAPTER   III. 

IMAGINARY    ROOTS    OF    EQ,TTATIONS. 

26.  Problem.  To  solve  a  binomial  equation,  and  re- 
duce all  its  imaginarij  roots  to  the  form  of%  1. 

Solution.     Let  the  equation  be 

Ax''—  M,  (47) 

in  which  A  and  31  are  real  or  imaginary,  and  a  a  positive 
integer.  When  (47)  is  divided  by  A  by  means  of  ^  18,  it  is 
reduced  to  the  form 

X'^  -ZZZ  777, 

in  which  m  is  of  the  form  of  ^  1.     Let  then 

m  z=z  r  (cos.  6  -\-  sin.  6.  \/ —  1),  (49) 

or  x«  =  r  (cos.  6  +  s'"-  ^-  V  —  !)•  (50) 

The  ath  root  of  (50)  is,  by  §  16, 

^  ^  A 

x=z/v/r.  (cos. {-sin. — .\/ — 1).  (51) 

a  a  '  ^     ' 

27.  Scliolium.  Since  0  has,  by  (9),  an  infinity  of  values, 
(51)  would  at  first  sight  appear  to  have  a  like  infinity  of  values. 
But,  by  (9), 

&  fl„  2  7Zrr 

—  =  —  dt  ,  (52) 

a  a  a  ^     ' 


14  IMAGINARY    QUANTITIES.  [b.  III.  CH.  III. 

Number  of  roots  of  a  binomial  equation. 

whence  the  values  of  x  are  identical,  when  they  correspond  to 
values  of  <^,  for  which  the  difference  of  the  values  of  n  is  equal 
to  a,  or  is  some  multiple  of  a.  Now,  by  subtracting  from  any 
value  of  n  the  greatest  multiple  of  a  contained  in  it,  a  remain- 
der is  obtained,  which  is  less  than  a.  The  number  of  differ- 
ent values  of  x  is,  therefore,  the  same  with  the  number  of  posi- 
tive integers  (zero  included)  which  are  less  than  a ;  that  is,  the 
number  of  values  of  x  or  the  number  of  roots  of  equation  (48) 
is  just  equal  to  a. 

28.  Corollary.    When  m  is  real  and   positivCj  (11) 
gives 

«,     /         2n-n:        .     2nTv  \ 

X  z=  \/m  i  cos. zb  sin.  — —  .V  —  1  I  >  (5^) 

in  which  the  double  sign  renders  it  unnecessary  to  no- 
tice those  values  of  n  which  exceed  the  half  of  a. 

29.  Corollary.     The  value  of  n 

n  =  0,  (54) 

reduces  (53)  to  its  real  positive  root 

a 

X  z=z  s/  m,  (55) 

30.  Corollary.    When  a  is  even  in  (53),  the  value  of  n 

nr=.\a,  '  (56) 

2w^  . 

gives  =  ^>  V^'y 

a 

X  =.  —  \/m.  (58) 


<5>  33.]  ROOTS    OF    Eq,UATIONS.  15 

Every  equation  has  a  root. 

31.   Corollary.    When  in  is  real  and  negative,  (12) 
gives 

«^         /         2/1  +  1           .     2n  +  l         ,      A      ,^^^ 
zi=V — wifcos. TTitsin. >T.\/ — II,    (o9) 

in  which  the  double  sign  renders  it  unnecessary  to  no- 
tice those  vaUies  of  n  which  exceed  the  half  of  a. 


32.  Corollary.  When  a  is  odd,  the  value  of  n 

—1 


n  =  %^.  (60) 


2n  +  l  '  ,^,, 

gives  Tc  ^::^  n^  (61 ) 


X  z=:  —  \/ — m.  (62) 

33.  Theorem.  Every  equation  has  at  least  one  real 
root  or  one  imaginary  root  of  the  form  (1). 

Proof.  Let  all  the  terms  of  the  equation  be  transposed  to 
its  first  member,  which  reduces  it  to  the  form 

/.x  =  0.  (63) 

Let  now  x^  be  any  real  or  imaginary  value  of  x,  for  which 
the  value  of  this  first  member  neither  vanishes,  nor  is  infinite, 
and  let  h  be  an  infinitesimal ;  let  also  dl.f.x^  be  the  first  dif- 
ferential coefficient  o{  f.x^  which  does  not  vanish  ;  and  (533 
of  Vol.  I.)  gives 

f{x^+h)  =r/.x,  +  ^^  3        ^^  d:.f.x^'  (64) 


16  IMAGINARY    QUANTITIES.  [b.  III.   CH.  IIT. 

Equations  whicli  have  finite  roots. 

Again,  let  i  be  «in  assumed  real  infinitesimal,  and  let  h  be 
determined  to  satisfy  the  assumed  binomial  equation 

r^r — -  '^"J.^o  =  -  if^x^.  (05) 

This  value  of  h,  being  substituted  in  (G4),  gives 

/.(/,  +70  =/.x,  -  z/:r,  =  (l-O/.^o  ;  (66) 

so  that  if  r  is  the  modulus  of/.a-Q,  that  of/.(.TQ  + /O  ^^>  ^7 
§11,  (1 — i)r,  and  therefore  less  than  that  o^  f.x^.  The  least 
possible  modulus  of  y.a;  is  then  less  than  r,  unless  r  is  zero  ; 
this  least  modulus  must  then  be  zero,  and  the  corresponding 
value  of  X  is  a  root  of  the  equation  (63). 

34.  Scholiiun.  The  preceding  argument  does  not 
exclude  infinity  from  being  the  root  of  the  given  equa- 
tion, so  that  the  following  is  a  convenient  statement  of 
the  above  theorem  ; 

Every  equation  has  at  least  one  finite  root  of  the  form 
(1),  wheiiy  after  it  is  reduced  to  the  form  (63),  it  does 
not  vanish  for  an  infinite  value  of  tlie  variable. 

35.  Corollary.  If  the  first  member  of  (63)  is  a  polynomial 
of  the  form 

xn  -]_  a  2-"-i  -\-  h  3;"-2  +  &c.,  (67) 

and  if  x'  is  a  root  of  the  equation,  this  polynomial  must  be 
divisible  by  x  —  x'  ]  and  the  quotient  must  be  a  polynomial  of 
the  (/^  —  l)st  degree,  which  must  be  divisible  by  a  similar 
factor  X  —  x",  and  so  on. 


<5)  37.]  ROOTS    OF    EQUATIONS.  17 


Tlie  conjugate  of  a  real  function. 


Hence  (67)  must  he  the  continued  product  of  n  dif- 
ferent factors  of  the  form  {.V  —  x^);  that  is,  the  equa- 
tion 

xnj^a  x«-i  +  h  2;"-2  -f  &,c.  =  0  (68) 

must  have  n  roots  of  the  form  (1),  whether  a,  b,  Sf'c.  be 
real  or  imaginary, 

36.  A  real  function  is  one,  which  has  real  values  for 
all  real  values  of  ihc  variable,  and  has  not  imaginary 
values,  unless  the  variable  is  imaginary. 

37.  Theorem.  The  conjugate  of  a  real  function  is  the 
same  function  of  the  conjugate  of  the  variable  ;  or, 
algebraically,  if 

P  +  QV-1=/.(P  +  2V-1),  (C9) 

where  /.  denotes  a  real  function,  then 

P-Qs/-\^f.{p-qs^-\).  (70) 

Proof.  The  function,  which  is  the  second  member  of  (69), 
may  be  developed  and  arranged  according  to  powers  of  \/ — 1. 
Let,  then,  the  aggregate  of  all  the  terms  which  are  independ- 
ent of  v' — 1)  'if'd  of  those  which  are  multiplied  by  even 
powers  of  \/ — 1  be  denoted  by  P ;  while  the  aggregate  of 
all  those  terms  which  are  multiplied  by  odd  powers  of  \/ —  1, 
is  denoted  by  Q'.  The  value  of  P  is  real,  and  remains  un- 
changed by  changing  v'  —  1  ^^  —  \/ —  1,  while  that  of  Q' 
is  reversed  ;  that  is,  the  value  of  the  function  is  changed 

from     P  +  Q'     to     P  —  Q'.  (71) 

2* 


18  IMAGINARY    QUANTITIES.  [b.  III.  CH.  III. 

Every  real  equation  has  at  least  two  roots. 

But  the  quotient  of  Q'  divided  by  \/ —  1,  containing  only 
even  powers  of  \/  —  1,  is  a  real  quantity,  which  may  be  de- 
noted by  Q,  that  is, 

Q'=zQx/— 1,  (72) 

P+Q'  =  P+QV  — 1;  (73) 

so  that  by  reversing  the  sign  of  \/ —  1,  (69)  is  changed  to 
(70). 

38.  Corollary,  When  Q  =  0,  (74) 
(69  and  70)  become 

-P=/-(i'  +  2V-i)=/.(i'-sV-i);      (75) 

that  is,  every  real  value  of  a  real  function  corresponds 
to  two  different  values  of  the  variable,  which  are  con- 
jugate to  each  other. 

39.  Corollary.  When  P  =  0,  (76) 
(75)  becomes 

0=/.  (p  +  5\/-i)=/.  (p-?Vi);       (77) 

that  is,  ivhen  the  function ^  which  is  the  first  meniher 
of  (63),  is  real^  tlie  conjugate  of  every  imaginary  root 
is  also  a  root  of  the  equation. 

40.  Corollary,  If  x'  is  a  root  of  the  equation  (68),  when 
a,  fc,  &c.  are  real,  and  if  x"  is  the  conjugate  of  x' ,  x"  is  also 
a  root  of  this  equation,  and  the  first  member  is  divisible  by 
the  product 

(z  —  X')  (x  —  x")  —  x^—  {x'  +  x'')  X  +  x'  x".  (78) 


<§.  40.]  ROOTS    OF    EQ,UATIONS.  19 

Number  of  real  factors  of  a  real  polynomial. 

If  r  is  the  modulus  of  z'  and  &  its  argument,  (8,  24,  and  36) 

give 

%'  -\-x"  —  "ir  COS.  6,     x'  x"  =  r2  ;  (79) 

whence  (78)  becomes  the  real  factor 

x2  —  2rx  cos.  6  +  r2  ;  (80) 

so  that  «???/  real  polynomial  of  the  form  (67)  is  the  con- 
tinued product  of  as  many  real  factors  of  the  form  x  —  x' 
as  the  equation  (68)  has  real  roots,  multiplied  by  the  co7i- 
tinued  product  of  half  as  many  real  quadratic  factors 
of  the  form  (80)  as  (68)  has  imaginary  roots. 

41.     Examples. 

1.  Decompose  z'^  —  h"^   into   a  continued  product  of  real 
factors  of  the  tirst  and  second  degree. 

Solution.     The  equation 

a;7  —  67  _  0,     or     x^  =  b^ , 

gives  in  (48)  m  =  b"^ ,     0  =  7; 

whence  (53)  becomes 

X  z=z  b  (cos.  f  n  TV  :iz  sin.  f  n  re .  a^  —  1 )  j 

which  becomes,  by  putting  saccessively  for  Ji  all  integers  less 
than  half  of  7, 

X  z=z  b, 

X  =1  b  (cos.  f  7r  i  sin.  f  tt  .  \/  —  1), 

X  z=z  b  (cos.  f-  ^  rt  sin.  f  ^  .  \/ —  1), 

X  z=:  b  (cos.  f  n  zh  sin.  f  ^  .  \/  —  1 ) ; 


20  IMAGINARY    Q,UANT1TIES.  [b.  III.   CH.  III. 

Decomposition  of  a  function  into  real  factors. 
SO  that,  by  (SO),  the  continued  product  is 

X'  —b'    ={x-^b)  {z2—2bx  COS.  f  .T-f&2) 

(x2_2  6  2;cos.  f -f +  62)  {x2—2bxcos.^^-\-b2). 

2.  Decompose  x^  -f~  ^*  ^^^^  ^  product  of  real  factors  of  the 
first  and  second  decree. 

Solution.     The  equation 

x^  -{-  b^  =1  0,     or     a-4  =  —.  6*, 

gives  in  (48) 

?«  =  —  b^,     —  ?n  =z  b^,     a  =z  ^; 

whence  (59)  becomes 

x  =  b  (cos.  i  (2  w  -]-  1 )  TT  i  sin.  ^  (2  71  +  1 )  rr .  V—  1 )  J 

which  becomes,  by  putting  successively  for  7i  all  integers  less 
than  2, 

X  =  b{c0S.irt-^sm.lrc,^—l)  =  b(h\/2:hiV^'V—l)y 

a:=6(cos.f  ^±sin.f:T.\/— 1)  =  ^(— W-=FiV-V— 1); 
so  that,  by  (80),  the  continued  product  is 
3c4_|_54_(3;2_26a;cos.-i-7r-|-62)(x2_263;cos.j7r  +  62) 

_  (3;2  —  ^  2  .  6  X  +  62)  (3;2  _|_  ^2  .  fe  X  +  62). 

3.  Decompose  x^ — 6*    into   a   continued  product  of  real 
factors  of  the  first  and  second  degree, 

Ans.     (x  — 6)  (x  +  6)  (x2  +  62). 

4.  Decompose  x^  +  6^    into  a  continued  product  of  real 
factors  of  the  first  and  second  degree. 

Ans.  (x+6)(x2— 26xcos.|7i:+62)(x2_26a;cos.f;T  +  62). 


$  40.]  ROOTS    OF    EQ,UATIONS.  21 

Decomposition  into  real  factors. 

5.  Decompose  x^ — h^   into   a  continued  product  of  real 
factors  of  the  first  and  second  decree. 

Arts,     {x—b)  {x+b)  (x2_|_5a;_j_52)  (^x^  —  bx  +  b^). 

6.  Decompose  x^  -\-  b^    into   a   continued  product  of  real 
factors  of  the  first  and  second  deorree. 

Ans.     (22-1-^3.6.2;+ 62)  (a;2-)-62)  {z2  —  ^S.bx-{-b2). 


22  IMAGINARY    Q,UANTITIES.  [b.  III.   CH.  IV. 

Imaginary  power. 


CHAPTER   IV. 

IMAGINARY     EXPONENTIAL    AND    LOGARITHMIC     FUNCTIONS. 

42.  Problem,  To  reduce  an  imaginary  power  of  a 
real  quantity  to  the  form  (1). 

Solution.  Let  the  exponent  of  the  power  be  ^-[-  B  a^ — 1, 
and  let  R  be  the  modulus  and  0  the  argument  of  this  power 
of  the  real  quantity  a,  that  is,  let 

a-^+^v-i  — jR(cos.  0  +  sin.  0.^—1).  (81) 

The  infinitesimal  power  i  of  this  equation  is  by  (28) 

«(^+5v-i)  —  Ri  (cos.  i  0  +  sin.  io.^—l).  (82) 

Hence  by  (418  of  Vol.  I.  and  §  22  of  Plane  Trig.) 

l-^i{A  +  B\/—l)\og.a—{l  +  i\og.R){l  +  i0^/—l) 

—  l+i{\og.R-{-0\/—l),  (83) 

Hence,  by  <5>  8,  and  using 

e  =  the  base  of  the  Neperian  logarithms,  (84) 

log.  R  —  A  log.  a  z=  log.  a-^,       R  —  a-^  ,  (85) 

B  log.  a  z=  0  =  log.  a^,      aB[z=  c©;  (86) 

which,  substituted  in  (81),  give 

a^-^B^-i  —  a-A  (cos.  B  log.  a  +  sin.  B  log.  a .  \/—\).  (87) 


<§>  47.]  LOGARITHMIC    FUNCTIONS.  23 

Imaginary  logarithm. 


43.  CuroUarj/.     When  ^  ==  0, 
(87)  becomes 

a^v/-i  —  COS.  B  log.  a  +  sin.  B  log.  a .  \/—  1.       (88) 

44.  Corollanj.     When  a  =:  e, 
(87  and  88)  become 

e-^+B^-i  —  e-^[cos.B  +  ^m.B,s/—\),  (89) 

gB./-i  _  eos.  B  +  sin.  B .  V—  1.  (90) 

45.  Corollary.    Reversing  the   sign  of  B,  (89  and  90)  be- 


come 


^A-B^f-i  —  e.^(cos.  B  —  sin.  S .  \/—  1),  (91) 

e-5v-i   z=  cos.  jB.  —  sin.  ^.  V— 1-  (92) 

46.  Problem.    To  reduce  the  logarithm  of  an  imagi- 
nary quantity  to  the  form  (1). 

Solution.     Let  r  be  the  modulus  and  <3  the  argument  of  the 
imaginary  quantity,  and  (90)  gives 

r{Q.os.&-\-s\n.&./^—\)  =  re(^^-^]  (93) 

the  logarithm  of  which  is 

log.  [r  (cos.  (3  -|-  sin.  6  .  /y/ — 1 )]  =:  log.  r  -\-  log.  c^  -^—"^ 

=:log.  r  +  6V— 1.     (94) 

47.  Corollary.     By  (4,  5,  and  94) 
log.(^+i?V— l)=logV(^2_[_2J2)_|_tan.[-i]^V— 1 

=  J  log.  (^2  _j_S2  )_ptan.[-i]^.  V— 1 ;  (95) 


24  IMAGINARY    QUANTITIES.  [b.  III.  CII.  IV. 


Number  of  the  logarithms  of  a  number. 


and  as  there  is  an  infinity  of  values  of 

a  =1  tan.[-i]  — , 
A 

every  quantity^  real  or  hnaginary^  has  an  infinity  of 
logarithms  J  of  w  J lich  there  is  never  more  than  one  real 
logaritlim^  and  that^  hy  ^  5,  only  ivhen  the  quantity  is 
real  and  positive. 

48.  Corollary,     By  §  5,  when  A  is  positive,  and 

B  —  Q, 
(95)  becomes 

log.  A  =  log.  A^'Znn  s/—  1,  (96) 

in  which  log.  A  of  the  second  member  is  the  real  value  of  this 
logarithm. 

49.  Corollary.     By  §  5,  when  A  is  negative  and 

5  =  0, 

(95)  becomes 

log.  A  —  log.  (_-  ,4)  ±  (2  n  +  1)  ^  V—  1.       (07) 

50.     Examples. 

1.  What  is  the  logarithm  of  J\/2(l  +\/— 1)? 

2.  What  is  the  logarithm  of  \/3  +  \/—  1  ? 

Ans.  log  2  +  (^  zh  2  n)  71  ^—  1. 


<§>  51.]  CIRCULAR    FUNCTIONS.  25 

Sine  and  cosine  of  imaginary  angles. 


CHAPTER   V. 

IMAGINARY    CIRCULAR    FUNCTIONS. 

51.  Problem.     To  reduce  the  sine  and  cosine  of  an 
imaginary  angle  to  the  form  ( I). 

Solution,     a.  Let  the  angle  be  B /s/ — 1,  which  being  sub- 
Btituted  for  B  in  (90  and  9^),  gives 

e-B  —  COS.  B  V—  I  +  sin.  B  sf—  1  .  V—  1,        (98) 
e^     =  COS.  2J^/— 1  —  sin   .BV— 1  .  V— 1-        (99) 

One  half  of  the  sum  of  (98  and  99)  is 

COS.  Bs/—\  —  l{c^-\-  e-B),  (100) 

One    half  of  the    difference   of  (98  and  99),   multiplied  bj 

iV —  1,  is 

sin.  ^V— 1  =  i{e^  —  e-^)\/— 1.  (101) 

b.  When  the  angle  is  A  -{-  B\/—},  (100  and  101)  give 
Bin,(^-f"^V' — l)  =  sin.^cos.  B/v/ — 1  -f-cos.  A  s\n.B\/ — 1 

=  ^sin.4(e^-[-e-^)  +  icos..4(e«  — e-^)>s/— I  ;   (102) 
eos.(  A-\-B\/ — I  )  =  cos.  A  cos.  B\/ — I  — sin.  ^4  sin.  By/ — 1 
=  icos.A(e^  +  e-^)  —  is\iuA(e^—e-^)^^l.    (103) 
3 


26  IMAGINARY    QUANTITIES.  [b.  III.  CH.  V. 

The  imaginary  angle,  whose  sine  exceeds  unity. 

52.  Problem.  To  reduce  the  imaginary  angle,  the 
absolute  value  of  whose  sine  is  greater  than  unity,  to 
the  form  (I). 

Solution.  Let  the  given  sine  of  the  angle  be  db  (1  +  «), 
and  let  the  required  angle  he  A  -{- B  \/ — 1  ;  it  is  evident 
from  (102)  that,  when  the  sine  of  the  angle  is  real, 

cos.^(e^  — e-^')  =  0;  (104) 

that  is,  either  e^  =  c-^,  (105) 

whence  e^^  =  1,     2  B  =  0,     i5  =  0  ;  (106) 

in  which  case  the   given   angle  is  real,  and  the  absolute  value 
of  its  sine  cannot  exceed  unity  ; 

or  cos.  J  =  0,      A  -=znn^  (107) 

sin.  yl  =  =b  1,  (108) 

whence,  by  (102  and  103),  (109) 
sin.(^  +  /^V— l)  =  sin.(7i7r  +  SV— 1) 

=:iM'^^  +  ^-^)  =  i(l+«),  (110) 

COS.(^  +  i^\/— l)=:C0S.(«7r4-i?V— 1) 
^zp^(e^— e-^)x/— 1 
_zp^(— .2a— «2)z==Fv/(2a+a2)V— 1.       (HI) 
The  sum  of  (HO),  and  (111)  multiplied  by  \/ — I,  is 

c^  =:  1  +  «  ±  V  (2  «  +  a^),  (112) 

whence 

JB  =  log.  [I  +fl±\/(2fl  +  fl2)] 

=  i  log.  [1  +  a  +  \/(2a  +  a^)l        (113) 
and  the  angle  is 

n^  d=  log.  [I  +  a+\/(2a  +  «2)].v'— 1.      (114) 


§  54.]  CIRCULAR    FUNCTIONS.  27 

Imaginary  circular  functions. 


53.     Examples. 

1.  Reduce  tang.  {A-\-  B  \/ — 1)  to  the  form  (I). 

2  sin.  2^  (e2^  — 6-2^)./— 1 

^'^^'    ^5_|_e-25  +2 COS. 2 ^ "^ c2^  +  e-2^  +  2 cos.24*  ^^^^^ 

2.  Reduce  tang.  S\/ — 1  to  the  form  (1). 

3.  Reduce  tang,  f— ^]  B  a^/ —  1  to  the  form  (1). 
Ans.     When  B  is  absolutely  less  than  unity,  it  is 

±w^  +  J[log.(l+^)  — log.  (1— J5)].V— 1.  (117) 

When  B  is  positive  and  greater  than  1,  it  is 
±(n+J).^  +  Hlog.(S+l)-log.(^-l)].x/-l.  (118) 

When  B  is  negative  and  less  than  —  1,  it  is 

± {n+ih+i[\og.-{l+B)-\og.(l-B)W-l'  (119) 
When  S  :=  =i=  I,  it  is 

^drOD.V— 1.  (120) 

54.  Equations  (100  and  101)  have  suggested  a  new  form  of 
notation  of  great  practical  value,  and  for  which  tables  have 
been  constructed,  similar  to  the  common  trigonometric  tables. 
It  consists  in  representing  — \/ — I  .  sin.  B  \/  —  1  and 
COS.  B  \/ — 1  by  Sin,^  and  Cos.  ^,  which  only  differ  in 
their  initial  capital  letters  from  the  common  trigonometric  no- 


28  IMAGINARY    QUANTITIES.  [b.  III.   CH.  T. 

Potential  functions. 

tatioii ;  this  notation  may  also  be  extended  to  the  other  trigo- 
nometric functions.  These  new  functions  are  called  potential 
functions.     We  have,  then, 

fi-      -r,  ,       ,     .      ^     #     ,        sin.  Bx/ — 1 

=  i{e^-e-^),  (121) 

Cos.B=icos,Bx/—l     =i{eB  +  e-B)^  (122) 

-„  „         Sin.  5         inner.  Ba^ — 1 

55.   Corollary.    The  differentiation  of  (121  -  123)  gives 
d,.  Sin.  5  =  J  (e^  +  e-B)  —  Cos.  B,  (124) 

rf,.  Cos.  J5  :=  4  (e^  —  e-^)  —  Sin.  ^,  (125) 

«//ran.  jB  z=  ^7^—,  =  7T-V^  =  Sec.2J5.  (126) 

'  C0S.2^V— 1  C0S.25  ^  ' 


56.     Examples. 

Demonstrate  the  following  equations. 

1.  C0S.2  B  —  Sin.2  B  =  1.  (127) 

Solution.     By  (121  and  122) 

C0S.2  B  =  1  (c25  +  2  +  C-2^) 

Sin.2  JB  =:  ^  (e25  __  2  _|_  g-sB) 
Hence         Cos.2  ^  _  sin.2  ^  —  1. 

2.  Sin.  {BdtiB')  =  S\n.B  Cos.B'  ±  Cos.  B  Sin.  B'     ( 128) 

3.  Cos.(JB ±  JB')  =  Cos.  B  Cos.^'i  Sin. B  Sin. 5'     (129) 

4.  Sin.(B  +  JB')  +  Sin.  {B—B')  =  2  Sin.£  Cos.^'    (130) 


^   54.]  CIRCULAR    FUNCTIONS.  29 

Potential  functions. 


5.  Sin.{B  +  B')  —  Sm.{B—B')  —  2Cos,BSm,B'  (131) 

6.  Cos.{B-{-B')  +  Cos.(B—B')  =  '2Cos.BCos.B'  (132) 

7.  Cos.{B+  B')  —  Co^B—B')  =  2  Sin.  ^  Sin.J3'  (133) 

Sin.  ^  +  Sin.^-_  Tang.  Hg  +  ^0  f.^.. 

'     Sin.  ^  —  ^m.B'  ~  Tang.  ^  (B  —  B)  ^       ' 

Cos  7? Cos  Tl' 

^-    Co:.B  +  Co:.B-  =  Tan.i(iJ+B')Tan.J(B-B')  (135) 

10.  Sin.  2B  z=2  Sin.  B  Cos.  5  (136) 

11.  Cos.  2  5  =  Cos.2  J5  +  Sin.25  (137) 

z=  1  +  2  Sin.2  B 
=  2  Cos.2  B  —  1 

12.  Sin.  ^jB=:  x/[|(Cos.  2  5  — 1)]  (138) 

13.  Cos.^B  =  ^[^{Cos.2B  +  l)]  (139) 

t4  rn         .  ^  ,  /  Cos.  2  5  —  1  \ 

14.  Ta„g.JB  =  v(c„i:2^+l)  (140) 

15.  Ta„g.(iJ±B')=.  5^^:^-^^^,  (141) 

1  zh  1  ang  jB  1  ang.  B'  ^       ' 

16.  Tang.  2B=              ^         .  (142) 

14-  1  ang. 2  5  ^       ' 

17.  d^.SinS-^}z=  {l  +  z^)-i  =  -^^±--.  (143) 

Solution.     Let         2:  ==  Sin.  5,  or  jB  =  Sin.[-i]  z 
Then  by  (124  and  127) 

ef,.^.x  =  Cos.jB  =  -v/(l  +  Sin.2 5)  =  ^(i  -j.  ^2) 
3* 


30  IMAGINARY    QUANTITIES.  [b.  III.  CH.  V. 

Potential  functions. 


and  by  (Vol.  I.  566) 


d,.  Sin.[-i]  X  :=  4,  B  = 


dcB'^         \/(l+^2) 


18.  rf..C03.[-i]x=  (.^-lH:zz^^J_^-      (144) 

19.  J,.Tang.[-i]x=-j-^.  (145) 


20.         Sin.  X  =  X  +  ^^-^  +  --^^^^  +  &c.   (146) 


21.        Cos.  X  =  1  +  --  +  :^^^-^-^  +  &c.  (147) 


§  59.]  REAL    ROOTS.  31 

Stern's  method  of  solving  numerical  equations. 


CHAPTER   VI. 

REAL    ROOTS    OF    NUMERICAL    EQUATIONS. 

57.  While  the  imaginary  roots  of  equations  are  of 
great  subsidiary  value  in  mathematical  investigations, 
and  frequently  admit  of  curious  and  interesting  inter- 
pretations in  physical  inquiries,  real  roots  are  the  prima- 
ry objects  of  attention,  and  methods  of  determining 
their  numerical  values  are  exceedingly  important  in 
practice.  Ster7i^s  method  is  the  simplest  which  has  yet 
been  published,  and  is  of  almost  universal  application. 

58.  If  the  values  of  a  given  function  and  of  its  suc- 
cessive differential  coefficients,  as  far  as  the  ?ah,  are 
found  for  a  given  value  a  of  the  variable  ;  and  if  the 
successive  signs  of  these  values  are  placed  after  each 
other,  the  row  of  signs  thus  formed  is,  in  this  chapter, 
called  the  nth  row  of  signs  (a),  or  simply  the  nth  row 
(a),  or  the  row  (a)  ;  any  pair  of  successive  signs  in  this 
row  is  called  a  permanence^  when  the  signs  are  alike, 
and  a  variation,  when  the  signs  are  unlike. 

59.  Theorem.  If  a  function  and  its  differential  co- 
efficients inferior  to  the  nVa  all  vanish,  but  the  ?/th  does 
not  vanish,  for  a  value  a  of  the  variable,  the  nth.  row 
of  signs   (a  +  i)j  *  being   an   infinitesimal,   consists 


32  IMAGINARY    QUANTITIES.  [b.  III.   CH.  VI. 

Signs  of  vanishing  functions. 

— 1 ^ — - 

wholly  of  permanences,  while  the  nih  row  (a  —  i)  con- 
sists wholly  of  variations. 

Proof.  It  follows  from  (Vol.  I.  533),  that  if/,  x  is  the  giv- 
en function 

/•  (« + 0  =  Y^ — ;;■  ''-■  J"-"-         ('^^) 

the  differential  coefficients  of  which,  taken  relatively  to  i,  are 

t.f.  ia  +  0  =  1T2T3  r:Wl)-  ''■^-  "  =  "-^^^'^' 

^  f(„A.i^-               '•-'                ..   .(n-\)f.(a+i) 
dl.f.  («  +  0  -  i.2.3...(„_2)-  ''-f-"- i ' 

&c.  &c.  &c.        (149) 

that  is,  all  the  terms  of  the  series 

/.(«  +  /),    d..f.{a  +  i),    dl.f.{a  +  i),&.c.         (150) 
have  the  same  sign. 

But  the  reversing  of  the  sign  of  i  in  these  equations  gives 

nf.  (a—i) 


dc'f-  {a—i)  = 


i 


ctl.f.  {a  —  i)  =  —^ ^-^^ S&c.  (151) 

that  is,  the  signs  of  any  two  successive  terms  in  the  series 

/.  (a  — 0,     d,.f.{a-~i),     e/.  («  — 0,  &c.       (J52) 
are  unlike,  and  the  terms  are  alternately  positive  and  negative. 

60.  Corolla?^.     If,  in  a  series  of  the  successive  dif- 
ferential coefficients  of  a  function  terminating  with  the 


<5»  61.]  REAL    ROOTS.  33 


Number  of  real  roots  between  given  limits. 


nth,  all  vanish  except  the  7i\h  for  a  vahie  a  of  the  vari- 
able, the  signs  of  this  series  will  in  the  row  of  signs 
(a  -}-  i)  constitute  a  series  of  permanences,  and  in  the 
row  (a  —  i),  a  series  of  variations. 

61.   Theorem,     If  the  first  member  of  the  equation 

f.x  =  0  (153) 

is  coiithmons  between  the  values  a  and  b  of  the  variable, 
a  being  greater  than  b.  if  tJte  nnniber  of  permanences 
in  the  nth  row  of  signs  (a)  exceeds  tJie  number  of  per- 
onaneuces  in  the  ntJi  row  (6),  and  if  the  excess  is  denot- 
ed by  V,  the  number  of  real  roots  of  (153),  ivhich  are 
included  between  a  and  b,  cannot  exceed  v. 

Proof.  For  while  the  value  of  x  varies  from  a  to  b,  a 
change  of  sign  can  occur  in  the  row  of  signs,  only  when  f  z, 
or  one  of  its  differential  coefficients,  or  a  series  of  them,  pass- 
es through  zero.  Now,  the  case  of  a  single  function  being 
included  in  that  of  a  series,  when  a  series  of  these  functions 
vanishes,  a  number  of  permanences  must,  by  ^^  59  and  60, 
be  lost,  equal  to  the  number  of  functions.  If,  then,  this  series 
begins  with  f  x,  as  it  must  when  the  variable  is  equal  to  a 
root  of  the  equation,  one  permanence,  at  least,  nmst  be  lost; 
that  is,  there  is  a  loss  of  one  or  more  permanences  in  the  row 
of  signs,  corresponding  to  every  real  root  of  the  equation. 

If  the  vanishing  series  does  not  begin  witli  f.  r,  and  con- 
sists of  an  even  number  of  functions,  the  sign  of  its  first  func- 
tion is,  by  (148-152),  the  same  with  that  of  the  function 
which  follows  the  series,  both  before  and  after  vanishing.  The 
relation  of  the  first  sign  of  the  series  to  the  sign  which  pre- 


34  IMAGINARY    (QUANTITIES.  [b.  III.  CH.  VI. 


Number  of  real  roots  of  an  equation. 


cedes  the  series  is,  therefore,  unchanged  ;  and  the  loss  of  per- 
manmrcs  is  exactly  equal  to  the  even  number  of  terms  of  the 
vanishing  series. 

If  the  vanishintr  series  consists  of  an  odd  number  of  func- 
tions, the  sign  of  its  first  function  is  reversed  when  it  vanishes. 
If,  therefore,  it  has,  before  it  vanishes,  the  same  sign  with  the 
preceding  function,  another  permanence  is  liere  lost,  which  is 
to  be  added  to  those  before  noticed.  But  if  it  has,  before  it 
vanislies,  the  opposite  sign  to  the  preceding  function,  a  new 
permanence  is  introduced,  when  it  vanishes,  which  is  to  be 
subtracted  from  the  number  of  the  others.  In  one  case,  there- 
fore, the  ivhole  number  of  lost  permanences  is  one  greater 
than  the  odd  number  of  terms  in  the  vanishing  series  ;  and, 
in  the  other  case,  it  is  one  less  than  this  number. 

In  any  case,  the  nuinber  of  lost  permanences  is,  at  least,  as 
great  as  the  number  of  real  roots  of  the  equation. 

62.  Corollary.  When  the  loss  of  permanences  does 
not  arise  from  a  real  root  of  the  equation,  the  number 
of  lost  permanences  is  even  ;  so  that  if  the  number  of 
lost  perrnanenres  is  odd,  that  is,  ifv  is  odd^  the  equation 
must  have  at  least  one  real  root  betioeen  a  and  b. 

63.  Problem.  To  find  all  the  real  roots  of  an  equa- 
tion. 

Solution .  Reduce  the  equation  to  the  form  (153), 
simplify  it  as  much  as  possible;  and  determine,  as 
nearly  as  possible  by  inspection,  those  limits  between 
which  the  different  real  roots  must  be,  if  there  are  any. 

Find  the  successive  differential  coefficients  of  the  first 


«5>  63.]  REAL    ROOTS.  35 

Stern's" method  of  finding  the  real  roots. 

member^  until  one  is  obtained  which  docs  not  vanish  be- 
tween two  limits  a  and  b,  between  which  there  may  be 
real  roots.     Let  this  be  the  nth  differoUial  coefficient. 

If^  then.,  a  being  greater  than  b,  the  nvmber  of  per- 
manences in  the  nth  roio  of  signs  (a)  is  the  same  ivith 
that  in  the  row  (b),  there  is  no  real  root  between  a  and 
b.  If  the  difference  between  the  number  of  'permanen- 
ces is  even,  the  question  of  a  real  root  between  a  and  b 
is  undecided  ;  and  if  this  difference  is  odd,  there  must 
be  such  a  root. 

Let,  then,  the  mth  differential  coefficient  be  the  hi^'h- 
cst  one,  of  which  the  sign  is  different  in  tlie  row  (a)  and 
in  the  roio  {b).      The  equatiofi 

f/,.-  +  i/.  xz=0,  (154) 

can  then  have  no  real  root  between  a  and  b,  luJtile  the 
equation 

d^rf.x-^,  (155) 

must  have  one,  which  can  be  found  by  the  process  gicen 
in  the  sequel  of  this  solution.  If  c  is  the  rnnt  of  (155), 
it  may  also  be  a  root  of  (153),  which  can  be  discovei^ed 
by  trial. 

However  this  may  be,  the  preceding  process  is  to  be 
repeated  for  the  limits  a  and  c  -\-  i,  i  being  an.  infinites- 
imal, and  also  for  the  limits  c-\-i,  and  b,  usitig  the 
mth  rov)  of  signs  instead  of  the  nth.  A  continuation 
of  the  process  '//nist  finally  lead  to  a  division  of  the  lim- 
its from  a  to  b,  into  sets  of  limits  so  norroiv,  that,  be- 


36  IMAGINARY    QUANTITIES.  [b.  II  I.  CH.  VI. 

Stern's  method  of  finding  numerical  roots. 


tween  each  set  there  con  only  be  one  real  root  of  ( 153) 
and  no  real  root  of  the  equation 

chfx  =  0.  (156) 

Lei  a'  and  h'  he  a  set  of  these  limits,  and  if  they  are 
far  apart,  substitute  for  x,  in  the  first  member  of[  153), 
different  numbers,  the  various  integers  for  instance^ 
between  a'  and  b' ,  until  one  is  found  which  does  not  dif- 
fer nincfi  from  the  required  root,  and  denote  this  first 
approximalio/i  to  tlie  root  by  x^.  Tlien,  if  the  exact 
root  is  XQ-\-h,  ive  have  by  (Vol.  I.  532) 

/.  (..„+/,)  =/.r„  +  h  ch.f.  {r„+«  A)  =0,  (157) 

whence,  by  neglecting  &h,  the  approximate  value  of  h 
is  obtained,  which  is 

h  =  -j^  (158) 

and  from  the  new  approximation  to  the  root  x^  +  A, 
which  is  thus  found,  a  neio  approximation,  can  be  ob- 
tained;  and  so  on,  to  any  required  degree  of  accuracy. 

64.  Corollary.  The  rate  of  approximation  can  be  readily 
determined  ;  for  if  two  successive  values  of  h  are  h  and  A' 
corresponding  to  a^  and  x^,  so  that 

rro  z:z  To  +  /i  (159) 

the  error  of  x^  differs  from  h  by  a  quantity  much  smaller  than 
A  ;  and  that  of  x^  is  nearly  equal  to  h' .     Now  suppose 

A<(Tvr  (160) 


<^  65].  REAL    ROOTS.  37 


Rate  of  approximation. 


and  we  have  by  (158)  and  by  Taylor's  theorem 
f.x'^=f{x,+h):=f,x,+  d,.f,x^,h-\.ldi.f,x,\h^+  &c.  (161) 
c/,/.2:;=rt/,./.Xo+&c.  (162) 

but  by  (158) 

f.x,  +  d,.f.x,.hz^O  (163) 

/.x;  =  J  ./?./.  2:0.  A2+&C.  (164) 

whence  neglecting  W^,  &c. 

If  now  we  find 

we  have,  neglecting  the  signs, 

/*'<(tVP+*  (167) 

and  therefore  if  one  approximation  is  accurate  to  s  places 
of  decimals,  the  next  will  be  accurate  ^0  2  s  +  A:  places, 

65.   Corollary.  Since  the  real  root  is  exactly 

^==^0  +  ^1 

we  have  ^0  =  ^  —  ^'»  (168) 

whence  by  (153  and  Vol.  I.  532) 

/.  x,  =  f(x  —  h)  =/.  X  —  h,d,,f  {x^&h) 

=z  —  hd,.f(x^6h),  (169) 

or  neglecting  &  k 

/.  Xo=  _  A  d,.f.  x={Xq^  x)  d,.f.  X.         (170) 
In  the  same  way  for  another  hypothesis  x^,  we  have 

/.  z;  r=  _  h'  d^.f  X  =  (x;  —  X)  d^.f  X.       (171) 


nr 


38  IMAGINARY    QUANTITIES.  [b.  III.  CH.  T. 


Rule  of  false  or  double  position. 


The  difference  of  (170)  and  (171)  is 

/.7,-/..r;  =  (zo-r;)c/,./.x  (172) 

and  the  quotient  of  (171)  by  (172),  is 

f'"    ^^'.^^I^  (173) 

which  is  identical  with  the  famous  rule  of  false,  or 
rule  of  double  positin??.,  iti  arithmetic;  and  this  admir- 
able rule,  the  principle  of  which  is  obviously  at  the 
foundation  of  all  higher  mathematics,  and  pervades  all 
practical  science  in  some  form  or  other,  is  sufficient  for 
obtaining,  with  ease  and  accuracy,  the  most  important 
numerical  results. 

66.     Examples. 

1.  Solve  the  equation 

xlog/ z— 100  —  0 

in  which  log.'  denotes  the  common  tabular  logarithms. 

Solution.  The  theory  of  logarithms  gives 
log.'  X  z=z  log.'  e  .  log.  X. 

Hence  if  f.x=^x  log.'  x  —  100 

d,.f.x  =  log.'z  +  log.'c 

The  value  of  d^^'f-  ^  is  positive  between  the  limits 

X  =  0,     and     x  zn  oo 
and  d^.f.x  is  negative  between  the  limits 

X  =  0,     and     x  =.  c~~^ 


^66.] 


REAL    ROOTS. 


39 


Solution  of  numerical  equations. 


at  both  which  limits  /'.  x  is  negative,  and  the  given  equation 
has  therefore  no  real  roots  between  these  limits.  But  d^.f.z 
is  positive  between  the  limits  x=:c~'  and  x  =  0,  at  which 
limits  f,  X  has  opposite  signs,  and  the  given  equation  has, 
therefore,  only  one  real  root,  which  is  between  these  limits. 

A  very  few  trials  show,   then,  that  the  root  is  not  far  from 
60,  for  which  value 

f.x  —  1,     d\.f.x—  1-78 +  0-43:=  2-21 

<^,./.xr=  -0072,      A:  =  2 

and  the  rest  of  the  calculation  may  be  arranged  as  in  the  fol- 
lowing form,  in  the  first  column  of  which  are  placed  the  suc- 
cessive values  of  y.x,  in  the  second  those  of  rf^'/^  ^»  sind  in 
the  third  those  of  x. 


7. 
0084860 


221 
219017 


60 
57 
569612 


2.  Solve  the  equation 

X  —  cos.  X  =:  0. 


Ans.     56-9612. 


Ans.     0-7391. 


3.  Solve  the  equation 

X  —  tang,  X  =  0. 

Ans.     There  are  an   infinity  of  roots,  one  being  contained 
between  each  set  of  limits 

n  n  and  (^  +  J)  tt 

in  which  any  integer  may  be  substituted  for  7i,   the  value  be- 
tween ^  and  J  ^  is     4*4934. 


^•'•f 


BOOK    IV. 


RESIDUAL   CALCULUS. 


4* 


A 


BOOK    IV. 

RESIDUAL     CALCULUS. 


CHAPTER   I. 


RESIDUATION.  "S^^' 


1.  For  every  finite  value  of  x,  which  satisfies  the  equation 

/x  =  00  ,  that  is,  ~  =  (/.x)-i  =  0,  (174) 

/•  ^ 

the  first  term  of  Taylor's  theorem  (Vol.  I.  442)  is  infinite,  and 
the  development  of  y.  (x  ~\-  h)  by  that  theorem  is  impossible. 
In  this  case,  if  i  is  an  infinitesimal,  f.  (x  -\-  i)  is  infinite  ; 
and  if  we  suppose  it  to  be  of  the  ?wth  order  of  infinity,  the 
expression 

imf,(x  +  i)  (175) 

is  of  the  order  zero,  and  is  usually  finite,  as  in  §  26  of  the 
Differential  Calculus.     The  quantity 

h^f.(x  +  h)  (176) 

may  then  be  developed,  by  MacLaurin's  Theorem  (445,  Vol.  I.), 
as  a  function  of  A,  and  the  result  will  be  of  ihe   orm 

h'"f,(x-\-h)  ==A  -{-  Bh-\-&.c. 
_j«  Qh^-^-{.Rhm-i  4-  Sh^^  +  r/i'^+i  +  ifcc.    (177) 


44  RESIDUAL    CALCULUS.  [b.  IV.   CH.  I. 


Residual.  To  residuate. 


which,  divided  by  7i'",  gives 

+  Q  A-2  4-  7?  A-i  4-  >S  4-  Th  +  &c.      (178) 

that  is,  f.{x  -\-  h)  can,  evcti  for  a  value  of  x  which 
satisfies  (174),  he  developed  in  a  series  consisting  of 
two  pa;  ^5,  one  of  luliich 

S  -\-  Th  ^  6cc.  (179) 

is,  like  Taylor^s  Theorem,  arranged  according  to  posi- 
tive and  ascending  powers  of  h,  and  the  other  part 

R  h-^  +  Q  /i-^  +  &c.  +  B  A-(— i)  -j-  A  A-'"    (ISO) 

is  arranged  according  to  negative  and  descending  pow- 
ers of  h. 

2.  The  coefficient  of  h~^,  in  the  development  of 
f.(^x-\-h)  by  the  preceding  method,  is  called  the  re- 
sidual  of/,  x^  and  vanishes  for  all  values  of  x^  except 
those  which  satisfy  (174). 

To  residuate  is  to  find  the  residual. 

3.  Problem.    2  o  residuate  a  given  function. 

Solution.  Let  f.  denote  the  given  function,  and  let  x^  be 
the  value  of  x  which  satisfies  (174).  Since  R,  which  is  the 
residual  of  ihi.-  function  by  (180),  is  the  coefficient  of  h™—^  in 
(177)  the  development  of  ]i"^f.{xQ  -\-  h)  by  MacLaurin's  The- 
orem ;  we  have  by  (445  of  Vol.  I.),  if  we  regard  h  as  the  vari- 
able, 


§  4.]  RESIDUATION.  45 

Method  of  residuating. 

provided  that  after  the  differentiation  we  put 

hz=  0. 

This  vanishing  of  h  may  be  effected  in  the  general  form,  bj 
substituting  for  h  the  infinitesimal  f,  which  gives 

^  =    1.2.3. ..(m-f)-  ^^^^^ 


4.    Examples. 

1.  To  residuate  the  function  (x  —  a)~^  (x —  b)~'^. 

Solution,  This  function  becomes  infinite  of  the  first  order, 
when 

x  z=i  a  -\-  i ; 

and  infinite  of  the  second  order,  when 

x  =  b  -\-  i. 
The  residual  which  corresponds  to  x  =.  a,  is,  then, 
'      i  (i)-i  («  —  6  +  i)-^  —  {a  —  b)-^  ; 
and  that  which  corresponds  to  x  :==  6,  is 

—  __(6_a)-2. 

1 

2.  To  residuate 


(x—a){x—b)  (z— c)3 
Ans,     The  residual  for  xziz  a,  is  {a — b)—^  (a — c)~^, 
that  for  x=b,  is  (b — a)-'^  (b—c)-^, 

that  for  x=c,  IS ^7^-7— r~ -• 


46  RESIDUAL    CALCULUS.  [b.  IV.   CH.  I. 


Residuation. 


3.  To  residuate  cosec.  2. 
Solution.     We  have 

cosec.  2  =  GO  , 
whenever  z=7i;r, 

71  being  an  integer,  and  the  residual  of  cosec.  z  is 

t  1 

t  cosec.  [n  jt -\- 1)^=- 


%$ 


sin.  {m^ -\- i)      cos.{nn^-j~i'j 
1 


COS.  n  n 


=  ±1. 


4.  To  residuate  tang.  x. 

Ans.     ±  1. 

5.  To  residuate  Cosec.  2.  Ans.     1, 

6.  To  residuate  (Cosec.  2)2.  Ans.     0. 

7.  To  residuate  2~^  cosec.  2.  .    .. 

-4/15.     When  2  =  0,     it  is  ^ ; 
%  when  2  z=  n;rr,  it  is  i  {n  7t)~', 

8.  To  residuate  x~^  cosec.  2. 

Ans.     When  2  =:  0,     it  is  0; 

when  2  =  n  TT,  it  is  i  (wti)"'. 

9.  To  residuate  — —  for  any  value  r^  of  z  which 

2 — z 

satisfies  the  equation 

/.  z  =  00  . 


^  4.]  RESIDUATION.  47 

Method  of  residuatins. 


%^SoJution.    Let  f.  {x^  -\-  i)  be  infinite  of  the  ?nth  order,  and 

let  f .  z  =r /.  z.  (z  —  3- J™,  (183) 

80  that  f.  (Xq  -j-  i)  may  be  of  the  zero  order,  and  the  required 
residual  is,  by  (182), 

d:-'.f.{x^+i).(x  —  x^  —  i)-' 
1.2.3....     (///.—  I) 

^  dT.T''r{^o+i)[(x-x^)-'+{x-T^)-H-{.(x^x^)-H2+&.c.] 
1.2.3        ....         (m  — 1) 

__  1  /d:-'  f-K  +  O 


1  .  2  .  3  .  .  .  .  »i  —  I  \         x 


( 


3", 


+^tSS^'  +  'E^?-'H— )(.»•) 


But  it  is  evident,  from  M  icLaurin's  Theorem,  that 

rf-\f.  (r.  +  O./" 


(185) 


J  .2.3...  (m— I) 
is  the  coefficient  of  i'"~'  in  the  development  of 

f.(.r,  +{).{"  (186) 

or,  dividing  by  «",  that  (IvSo)  is  the  coefficient  of  {"'-"-'^  in  the 
development  of  f .  [x^  -\-  i).     Hence,  by  this  ilieorem, 

d^fSJ^^^+i)J^  _  f/— -^ .  f .  (r^  +  0  _  r/r"-^f.(xj 
1.2.3...(m-l)""  1:2.3....  (m—/i—l)       1.2.3...(77^-/^-l)^      ^ 

which,  substituted  in  (184),  gives,  for  the  required  residual, 
1.2.3...  (m-l)'x—XQ~  1.2.3...  (/«-2)*(x—2q)2~        

"T"     1.2      '(x— Jo)"'-2    '     (Z—Xj"'-^    '"(X  — Zq)-        ^  ^ 


48  RESIDUAL    CALCULUS.  [b.  IV.  CH.  I. 

Integral  residual. 

The  value  of  f.x^  is  found   by  the   equation  (183),  which  by 
(521  of  Vol.  I.)  gives 

_  d-{z-x,r  _  1-2    3>-  :^^         ,189^ 

10.  To  residuate  the  preceding  example,  when 

z^  -\-  ab 


f.^  = 


{z—a)  (z—l>)^' 


a^J^ab       1 

Ans.  When  z^^a.  the  residual  is -—  . ; 

(a — by'    X — a 

.    .    b2—2nb—a2       1  b^-^-nb  1 

when  z==b,  it  is — —  . -\- 


(^b—af       'x—b^    b—a    '  {x  —  by' 

11.  To  residuate  example  9,  when 

f  .z  z=.  cosec.  z. 

Ans.     When  2  =  w  tt,  the  residual  is  ri= 


X  —  n  7t 


5.  The  ijiteg?^al  residual  o(  3.  function  between  cer- 
tain limits  is  the  sum  of  all  its  residuals  contained  be- 
tween those  limits;  and  the  total  residual  is  the  sum  of 
all  its  residuals. 

To  residuate  from  one  value  of  a.  variable  to  another 
is  to  find  the  integral  residual  between  these  values  of 
the  variable  ;  and  to  residuate  totally  is  to  find  the  total 
residual. 

a.  The  total   residual  is  indicated  by  the  sign  ^,  and  the 


^  5.]  RESIDUATION.  49 

Notation. 

integral  residual  is  denoted   by  the  same  sign  with  letters  an- 
nexed to  it,  to  show  the  limits  of  the  residuation  ;   thus 

t(/-^)  (190) 

is  the  total  residual  of /".  x  ;  while 

^^■(/•^)  (191) 

is  the  integral  residual  ofjT.  x  from  the  limit 
X  =  Xq      to      X  =  Xj. 

b.  The  residuation  is  often  limited  to  those  values  of  the 
variable,  which  render  one  of  the  terms  or  factors  of  the  given 
function  infinite,  as  in  Example  9  of  the  preceding  section  ; 
and  this  is  indicated  by  placing,  in  double  parentheses,  the 
factor  which  is  thus  regarded  exclusively  of  the  other  factors. 

Thus  £•((/•  ^))-(/'-^)  (192) 

indicates  the   residual   of  (y.  x)  (/"'.  x)  with  regard   to  those 
values  of  x,  which  render  f.  x  infinite.     In  this  way 

£-((/-^))  (193) 

should  be  usgd  instead  of  (190)  to  denote  the  total  residual  of 
f.  X.     In  the  same  way 

denotes  the  simple  residual  of 

/.x.(x-r„)^ 

X  — Xq 

for  the  value  of  x,  x  z=z  x^. 

5 


50  RESIDUAL    CALCULUS.  [b.  IV.  CH.  I. 

Notation. 

6.  The  variable  in  (101)  may  be  itself  a  function  of  other 
variables,  as  y,  z,  &c. ;  and  the  residuation  may  be  sought 
between  the  limiting  values  of  y 

y  —  Vo  ^"^  y  —  yxy 

and  those  of  z 

z  =  Zq   and  2  =  2;^,  &c. 

and  this  may  be  expressed  by  the  form 

vy  =  y,.  ^  =  ^1,  f^-.((/.:r)),  (195) 

or  more  simply 

it  being  conventional  in  what  order  the  limits  are  placed. 

7.   Corollary.  The  preceding  notation  gives  at  once,  if  x'  is 
a  value  of  x  between  x^  and  x^, 


ryl 


l7-i(f-^))  =  ei  •{{/■^))  + 17 ((/■-))•    (19-) 


8.  Scholium,     If  x'  is  a  root  of  the  equation 

/.xzzroo;  *  (198) 

the  value  of  the  corresponding  residual  should  be  equally  di- 
vided between  the  two  terms  of  the  second  members  of  (197), 
that  is,  when  one  of  the  limits  of  (191)  is  a  root  of  (198),  one 
half  of  the  corresponding  residual  should  be  included  in  the 
expression  (191). 

9.  Corollary.    If,  in  (19G),   there   are  only  two  variables  y 
and  z,  and  if  y  is  taken  to  denote  the  real  term  of  x  reduced 


<5.    10.]  RESIDUATION.  61 

Residual  of  differential. 

to  the  form  (1),  and  z  ilie  real  factor  of  the  imaginary  term, 
(196)  will  denote  the  integral  residual  for  all  values  of  2", 
whose  real  terms  are  included  between  y^  and  ?/j,  and  the 
real  factors  of  whose  imaginary  terms  are  included  between 
Zq   and  z^. 

10.  Corollary.  It  is  evident  from  (182)  and  <5»  5? 
that  the  residual  is  a  linear  function  ;  and  found,  as  it 
is,  by  differentiation,  it  must  by  <§>  52  of  B.  II.  be  free 
relatively  to  any  other  linear  function^  such  as  differ- 
ence,  differential^  &c. 

Thus,  if  the  residuation  is  taken  relatively  to  .r,  we 
have 

L.{{d^-.f.{x,z)))  =  d^-.C{(f.(x,z))),  (199) 


52  RESIDUAL    CALCULUS.  [b.  IV.   CH.  II. 


Development  of  a  function,  wJiicli  has  infinite  values. 


CHAPTER   II. 

DEV^ELOPMENT    OF    FUNCTIONS,    WHICH    HAVE    INFINITE 

VALUES. 

11.  Problem,  To  develop  a  funclion  which  has  in- 
finite values  corresponding  to  finite  values  of  its  vari- 
able^ in  a  fortn  which  may  be  tised  for  all  values  of  its 
variable. 

Solution,  het  f.x  be  tlie  given  function,  and  let  x^  be  a 
value  for  which  it  becomes  infinite,  so  that,  if  i  is  an  infini- 
tesimal, f{xQ  -\-i)  is  infinite  of  the  mih  order.     Then  if  we 

put  f:.xz=fx.{x  —  x^)'^;  (200) 

we  have  f .  Xq  finite,  and  (200)  can  be  developed  according  to 
powers  of  x  —  x^.     We  have,  by  Taylor's  Theorem, 

d""-^  f   X 
whence,  by  (200), 

f  r  //  f   r  d"^-^    f  X  1 

•^         (x-a:J-^(x-xJ'«-i^  ^1.2.3...(wz-l)  x—x^ 

d'^  f  X  d"'+^  f   X 

^1.2.3... 7»^1.2.3.. .. (m+1)  ^         °''~  '^       ^ 

Now  the  upper  line  of  the  second  member  of  (202)  consists  of 


<5>   11.]  DEVELOPMENT    OF    FUNCTIONS.  53 

Function,  which  is  always  finite,  when  the  variable  is  so. 


terms  divided  by  different  powers  %  —  Xq,  all  of  which  are 
finite,  unless 

%=x„  (203) 

in  which  case  they  are  infinite  ;  while  the  lower  line  is  a  func- 
tion of  X,  which  is  finite  in  this  case.  We  will  denote  the 
upper  line  by  X^  and  the  lower  line  by  Y^^  \  and  X^  is,  by 
(188),  the  residual  of 

-^-^^  (204) 

X  —  z 

when  z  =  x^.  (205) 

If,  then,  we  denote   by  Z^   all  the  other  residuals  of  (204), 
when  jf.z  is  infinite;  we  have,  for  the  total  residual  of  (204), 

iS^l^-^X^^Z,.  (206) 

But  by  (202)  /.  z  =  Xo  +  Fo  J  (207) 

and  therefore  /.  x  —  ^.  ^^^^-^  =  ^o  —  ^o-     (^08) 

Now  Fq  and  Z^  are  both  such  functions  x)f  x  that  they  are 
finite  when 

X  =  Xq  ;  (209) 

that  is,  the  first  member  of  (208)  is  a  function  of  a;,  which  is 
finite  for  every  finite  value  of  t,  such  as  (209),  for  which  f»% 
is  infinite,  and  if  we  denote  this  function  by  w.x^  we  have 

f-—l^T^ --.■-•  (210) 

Hence  the  second  term  of  (210)  is  finite  for  all  finite  values  of 
X  for  which  f.  x  is  finite  ;  and,   therefore,  -cs   x  must  be  Jinite 
for  every  Jinite  value  of  x. 

5* 


54  RESIDUAL    CALCULUS.  [b.  IV.  CH.  11. 


Development  of  a  function,  which  has  infinite  values. 
Hence  in  the  equation 

/..=  £/i4^  +  ...,  (211) 

the  first  term  of  the  second  member  consists,  as  in  (188),  of  a 
combination  of  terms  arranged  according  to  the  negative  pow- 
ers of  X  —  Xq,  X  —  a; J,  &,c.,  while  or .  x  is  always  finite,  and 
can  usually  be  developed  according  to  powers  of  x  by  Taylor's 
Theorem,  or  by  some  other  simple  process. 

12.  CoroUari/.  When  the  modulus  of  x  is  infinite,  the  first 
term  of  the  second  member  of  (211)  vanishes,  and  (211)  be- 
comes 

f.co=  cr.x.  (212) 

13.  Corollary.  When  the  first  member  of  (212)  is  finite  for 
all  values  of  the  argument  of  z,  cr.  x  is  always  finite.  But  it 
has  been  shown,  in  ^  81  of  B.  III.,  that  the  equation 

-i-rrO,     or     tjr.x  =  00,  (213) 

uf  X 

is  always  possible,  unless  cr.z  is  constant,  that  is,  independent 
of  x;  and,  therefore,  if  we  put 

/.  O)  =  P;  (214) 

we  have 

iiif  x^F,  (215) 

and  in  this  case 

f.^^lSSl-llj^F.  (216) 


^   16.]  DEVELOPMENT    OF    FUNCTIONS.  5$ 

Development  of  a  rational  fraction. 


14.  Corollary.  When  f.x  is  a  rational  fraction,  zj.x  is 
also  a  similar  rational  fraction,  because  the  second  term  of 
(210)  consists  of  the  sum  of  such  fractions.  But  cr.  x  cannot 
have  an  entire  polynoiniai  for  its  denominator,  because  such  a 
denominator  would  vanish  for  finite  values  of  x,  and  cj.  x  would 
become  infinite.  Its  denominator  must  then  be  constant;  that 
is,  cr.  X  must  be  an  integral  polynomial. 

15.  Corollary.  If,  in  the  preceding  corollary,  the  degree  of 
the  numerator  of/",  x  is  greater  than  that  of  its  denominator, 
this  function  is  infinite  when  its  variable  is  infinite;  but  if  the 
degree  of  the   numerator    is  equal   to  that  of  the  denominator, 

f,x  is  finite  when  its  variable  is  infinite  ;  but  if  the  degree  of 
the  numerator  is  less  than  that  of  the  denominator,  f,  x  van- 
ishes when  its  variable  is  infinite.     For  if  the  function  is 

^x"+/>x"-i  4-  &c. 
J  -  —  a'x^'+  bx"  -'4-  6lc.  ^'•^  '  ^ 

we  have  /.  x  =  -— — {--  =  —  (go  )"-«'  (218) 

which  is  infinite,  when     n  >  »', 

finite  and  =  -y  i=  i^,   when     n  z=  n',  (219) 

zero,  when     n  ^  n'. 

The  polynotTiial  uj.x  is,   therefore,  reduced  to  a  constant  in 
the  second  case,  and  to  zero  in  the  third  case. 

16.  Corollary.  The  easiest  way  of  finding  zu .  x  m  the  case 
of  §  14,  is  to  reduce  the  given  fraction  by  division  to  a  mixed 
expression,  consisting  of  an  integral  polynomial,  and  a  fraction 
in  which  the  degree  of  the  numerator  is  less  than  that  of  the 


56  RESIDUAL    CALCULUS.  [b.  IV.   CH.  II. 

Development  of  cosecant. 

denominator.  For  this  last  fraction  can,  by  the  preceding 
corollary,  furnish  no  part  of  the  polynomial  et.  ■x,  ^vhich  must, 
therefore,  be  the  same  with  the  polynomial  thus  obtained  by 
division. 


17.     Examples. 

1,  Develop  (sin.  a)— ^  by  the  preceding  principles. 

Solution.    The  general  expression  for  the  root  of  the  equa- 
tion 

(sin.  x)-i  =  00  ,  (220) 

is  x  =  ±/i-^,  (221) 

in  which  7i  is  any  integer  at  pleasure;  and  the  corresponding 
value  of  the  residual  of 


(sin.  z) 
X  —  z 

is,  by  Ex.  9,  §  4,  if  we  put 

1 

i.z  = 


— 1 


d^ .  sin.  z         cos.  z 
1  1 


cos,  lire     x^nrc 

SO  that  by  (21G) 


(222) 


1  1  1  1,1,1 

cosec.  X  =  -, = j r— _  H -- 

sin.'x       X       x-\-n       X — n       x-f-'z/v      X — 4  TV 


<5>  17.]  DEVELOPMENT    OF    FUNCTIONS.  57 


Development  of  secants. 


2.  Develop  sec.  x  by  the  preceding  principles. 

Anz.  sec.  2  =  4  71 1— — -—^ —      .,    .  -7^  +  ^^^"^ — v^ — &.c.  J 

(224) 

3.  Develop  {e  +  c-^)-i  =  J  Sec.  2;. 

Solution,     Let         a;  z=  ?/  +  ~  V —  1»  (225) 

and  we  have,  by  (89), 

e^   z=z  cv  {co%.  z -\- A^ — l.sin.  %),  (226) 

€—''=:  e—y{cos.  z  —  V  —  1 .  sin.  z).  (227) 

Hence  the  equation  e^-|~  ^~"  ^=  ^^  (228) 

involves  the  two         (e^  +  e~y)  cos.  z  z=  0,  (229) 

(ey__e-y)  sin.  ;:i  =z  0.  (230) 

Hence,     cos.^z=0,     e'^  —  e-y,     or     e^^—l,     y=:0;(231) 

z  =d^(n  +  i)n,  (232) 

and  the  root  of  (228)  is 

^{n  +  i)n^-l,  (233) 


7: 


If,  now,  we  take 


f-=^^e---^=^--^''  (~^'') 


we  have,  by  (90  and  92), 

1 
2x7—1 

and  the  corresponding  residual  of 

X  —  z 


{.r,  =  ±  ;r-7—i  ■'  i~^^) 


(236) 


5S  RESIDUAL    CALCULUS.  [b.  IV.  Oil.  II. 

Development  of  a  rational  fraction. 

11  1 

^^    ^o     /       1-    -^/.   ,    1-,—/— 1=^^ 


'2V— l^=P(^^+=^)-'V— 1  -^/V— i±(2w+l).T 

(237) 
we  have,  then, 

-L-  =  ( ' L__\ 

4.  Develop     (c""  —  e— ^)— ^  =z  J  Cosec.  x. 

3-5    4-    1 


Solution.     Since 

(.i-        1)2  (x  +  2)  z=  x3        3z  +  2, 

(240) 

re  have,  by  division, 

*'+!        -t-  1  3  1   -2^^+9^-5 

(241) 

(^_,)-2(,+2)--      1       1    (x_I)2(x  +  2)- 

Now  by  Ex.  9  of  §  4 

^  \\(z-\y^(z^'2))  fx-z       3(x-l)2 


I         ^^ ^^        .  ^24^^^ 

"T-9(x  — 1)       9(z  +  2)'  ^     "^ 


<5>   17.]  DEVELOPMENT    OF    FUNCTIONS.  59 

Development  of  rational  fractions, 
whence,  by  (216), 


^^  +  '  -.3+3+         2 


{x  —  \)2  (x-^2)  '         '    3{x—  1) 


2 


^      ^^      -      ^^  m3^ 

"^9(x— 1)       9(x  +  2)'  ^     ^ 

/*.  a; 

6.  Develop — ^— — ^^ ,  in  which  x^.,  x,, 

{x—Xo){x—x^){x  —  X2).... 

&/C.  are  all  unequal,   and   the  values  of  x,  which  rentier  f.  x 

infinite,  are  to  be  necrlected. 

Solution.    We  have  at  once 
Zlf —   r /l^ 1_      (244) 

(X-X,){X-X^)...  C     (((^_3;^)(^_:,J...))       ^_^  V  / 


(^•O  — ^^l)('0  — ^2)---     ^—^ 


f.x  1 

{x^—Xq){x^—X2)(x^—x.J...    x  —  x^ 


7.  Develop 


(x+l)(x_2) 


^"'*       3C^l)"*"3(x— 2)' 

a  Develop    ^-^---^^. 

2  2 


(a;_2)2       3(x— 2)    '    3(x+  1) 


BOOK   V. 


INTEGRAL   CALCULUS. 


BOOK    V. 


INTEGRAL     CALCULUS. 


CHAPTER   I. 

INTEGRATION. 


1.  The  iyitegral  of  a  given  differential  is  the  func- 
tion of  which  It  is^th-  diifereiitial ;  and /Ac  integral  of  a 
given  finite  function  is  the  function  of  which  it  is  the 
differential  coefficient. 

To  integrate  is  to  find  the  integral.  The  sign  of  integra- 
tion is  /. ;  thus 

f.d,x  =  x,    f.d.fx  —  f.x] 

f.d,.x  =  x,    f,d,.f.x=fx;  (245) 

f'di.x  =  x,    f:d:.fx=fx,&.c.  (246) 

2.  Corollary.    Since  we  have 

d,i{x  +  a)=zd,.fx,  (247) 

for  all  values  of  a,  it  follows  that 

/.c/,./.xr=/.x  +  fl,  (248) 

that  is,  the  integral  of  a  function  may  have  an  arhitra- 


64  INTEGRAL    CALCULUS.  [b.  V.  CH.  I. 

Increase  or  decrease  of  arbitrary  constant. 

ry  constant  added  to  it,  and  in  this  form  the  integral  is 
said  to  be  complete, 

3.  Corollary.  Any  constant  may  then  be  added  to,  or  sub- 
tracted from  the  incomplete  integral,  and  the  form  of  the  in- 
tegral may  often  be  changed  by  this  process. 

4.  Corollary.  If  the  integral  contains  a  term  of  the  form 

log./  ^, 

this  term  may  be  changed,  by  the  addition  of  a  constant,  to  the 
form 

log./.  X  +  log.  a  =  log.  {af.  x).  (249) 

5.  Corollary.  If  the  integral  contains  a  term  of  the  form 

sin.t— ^]  X, 

this  term  may  be  changed,  by  the  addition  of  a  constant,  to  the 
form 

sin.C-i] 2  —  J^rz:  —  (Jtt —  sin.[-i]  x)z=i  —  cos.[-i]  x  (250) 

or  it  may  be  changed  into 

coseJ-^]  -,  or  into  cosj— i]/\/(l-x~)  or  into  -  s'mS~^^\/{l-x^), 

In  the  same  way,  terms  of  the  form 

cos.t— ^]  2:,     tan.[— ^]x,     cot.f— ^]  z,     sect— ^3  x,  &,c. 

may  be  changed  into 

—  sin.[— ^]  x,     cot.t— 1]  X,     — tan.-— ^]  z,     — cosec.t— ^3  2,  &:c. 

or  into 

1.1  1  1  ' 

sect— ^-' -,   — tan.'-^^ -,    tant— ^1-,    cos.t— ^]-,  &c. 

Z  XXX 


«J  7.]  INTEGRATION.  65 


Number  of  arbitrary  constants.     Definite  integral. 


or  into  (251) 

6.  Corollary.  Since  every  integration  introduces  an 
arbitrary  constant,  the  nunib(3r  of  arbitrary  constants 
in  a  complete  integral  must  be  equal  to  the  number  of 
integrations. 

7.  Corollary.  The  difference  between  the  two  values 
of  an  integral,  which  correspond  to  two  values  of  its 
variable,  is  called  the  definite  integral  from  one  value 
to  the  other  value  of  the  variable. 

Thus  if  Xq  and  Xj  are  the  limiting  values  of  the  variable, 
the  integral  of  cl^.f.  x  from  x^  to  r^  is,  by  (248), 

(/.  ^1  +  «)  -  (/.  ^0  +  «)  =/•  ^2  -/.  ^0  ;         (252) 
and  it  is  written 

•X. 


f 


Kd,.f.x=f.x^.  (252) 


The  definite  integral  is,  therefore,  independent  of  the  value  of 
the  arbifrary  constant ;  but  the  places  of  the  arbitrary  con- 
stant and  the  variable  are  supplied  by  regarding  one  of  the 
limits  as  arbitrary  and  the  other  as  variable,  thus 


J  ^( 


,d,.f.x=f,x^f.x^,  (254) 

which  gives,  by  (248), 

a  —  — /.  3-Q,  (255) 

6* 


66  INTEGRAL    CALCULUS.  [b.  V.  CH.  I. 

Integrals  are  linear  functions. 

8.  Corollary.     Since 

/^o.rf../.x=/.x„-/.x.,  (256) 

we  have,  obviously, 

fi:=-fi:-  (257) 

9.  Corollary.  Equation  (246)  shows  that  integration  may 
be  regarded  as  negative  differentiation,  that  is, 

^"^Z  (258) 

10.  Corollary.  It  is  evident,  from  B.  II.  '§^^  51  and 
62j  that  integrals  are  linear  functions^  which  are  free 
relatively  to  all  other  linear  functions. 

Thus  we  have  f.af.x  —  aj.f  x.  (259) 

11.  Corollary.  Differ eritials,  residuals ^  a7id  integrals 
are  functions  which  are  relatively  free. 

12.  Corollajy.  When  a  function  can  be  separated 
into  parts  connected  by  the  signs  +  or  — ,  the  integral 
of  the  ivhole  function  is  the  algebraic  sum  of  the  partial 
integrals. 

This  method  of  integration  might  naturally  be  called  inte- 
gration by  parts,  but  the  following  is  a  particular  case  of  it,  to 
which  this  designation  has  been  applied  technically. 

13.  If  u  and  v  are  functions  of  a  variable,  we  have  (Vol.  I. 

468) 

d^.uv  z=i  udg.v  -\-  V  df.u,  (260) 


<5)   16.]  INTEGRATION.  67 

Integration  by  parts. 

whence  ud,,.v  ^n  d^.uv  —  vd^.u,  (201) 

and  by  integration 

f,ud^.v  z=  uv  —  f.vd,.u;  (262) 

and  when  a  given  differential  coefficient  can  be  sepa- 
rated into  two  factors,  one  of  which,  d^.  v,  has  a  known 
integral,  the  integration  can  often  be  effected  by  the 
aid  of  (262) ;  and  the  application  of  this  formula  is 
called  integration  hy  parts. 

14.  Theorem,  A  definite  integral,  which  is  taken  be- 
tween limits  differing  by  a  quantity  equal  to  the  differ- 
ential of  the  variable,  is  equal  to  the  differential  of  the 
integral. 

Proof.     For  the  equation  (252)  becomes,  when 

3-0  =  2;,         a;  J  r=  z  +  c?  X,  (263) 

by  (Vol.  I.  421) 

fl^'\d,.f.x=:f.{x-\.dx)^f.x^d.f.x.       (264) 

15.  Theorem.  \i  x^^  x^^  x^^  ,  ,  ,  .  x^  are  successive 
values  of  x,  a  definite  integral  from  Xq  to  ^,„  is  equal 
to  the  algebraic  sum  of  the  corresponding  definite  inte- 
grals from  Xq  to  x^j  from  x^  to  x^^  6oc. 

Proof.    We  evidently  have 

/.^n— /.2:o  =  (/.2;i  — /.Xq)  +(/.Z2  — /.2;  J 

+  (/.^3-/.^2)  +  &'C.       (265) 

16.  Corollary,  Hence  if  Xq^  x^^  x^^  6cc.  differ  by  dx, 


6S  INTEGRAL    CALCULUS.  [b.  V.  CH.  I. 

Change  of  variable. 

the  definite  integral  from  x^  to  x„  is  equal  to  the  al- 
gebraic sum  of  all  the  corresponding  differentials  from 
Xq  to  x,i,  taken  at  intervals  equal  to  dx. 

17.  Scholuim.  Propositions  14  and  16  require  that  the  inte- 
gral be  a  continuous  function  between  the  limits,  and  particu- 
lar caution  must  be  observed  to  exclude  those  cases,  in  which 
the  value  of  the  integral  varies  from  positive  to  negative,  or 
the  reverse,  by  passing  through  infinity,  so  as  suddenly  to  vary 
from  positive  to  negative  infinity,  or  the  reverse. 

18.  Theorem.    If  we  have  the  equation 

f.f.x^F.x  (266) 

and  if  we  substitute  for  x  any  function  at  pleasure,  as 
<f  .  X,  we  shall  have 

f.f.{cp.x).d,.cp.x  =  F.<p.x.  (267) 

Proof.     For  (266)  gives  by  differentiation 

d,.F.x=f.x,  (268) 

and  putting  x  z=z  ip.  y  (269) 

we  have,  by  (Vol.  I.  566), 

4,  F.if.y  =zf,{if.y).d,.ip.y,  (270) 

and  by  integration 

F.cp.y=:f.f.{<p.y).d,.cp.y,  (271) 

which  is  the  same  with  (267),  changing  y  to  i. 

10.  Corollary.     When     xz=z  (f.y  (272) 

we  have  f'f-^^S'f-i'P-y)'  d,.(p.y.  (273) 


<^  20.]  RATIONAL    FUNCTIONS.  69 

Integration  of  algebraic  monomials. 


CHAPTER   II. 

INTEGRATION    OF    RATIONAL    FUNCTIONS. 

20.  Problem.   To  integrate  an  algebraic  monomial. 
Solution,    First.    If  the  algebraic  monomial  is 

Gz",  (274) 

in  which  n  differs  from  —  1 ,  the  substitution  in  2G2  of 

w  z=  a  X",     u  z=  J,  ,  (275) 

d^.uzzzn  a  x"-\     d^v  z=z  \j  (276) 

gives  f.ax'^zzia z^+i — /. nax''=  a  x"+i  —  nf.  ax""  ;  (277) 
whence,  by  transposition  and  division, 

nf.  ax^+Z.  ax"=  (7i  +  l)f.ax^  =  ax«+i,  (278) 

that  is,  the  integral  of  the  monomial  is  found ^  by  in- 
creasing the  exponent  of  the  variable  by  unity,  and 
dividing  by  the  exponent  thus  increased. 

Secondly.  An  arbitrary  constant   should   be   added  to  (279) 
for  the  complete  integral,  and  we  have  as  in  (25-1) 

A.„,.^il£l±f^  (280) 

J  x^  n  4-  1 


70  INTEGRAL    CALCULUS.  [b.  V.  CH.  H. 


Integration  of  algebraic  monomials. 


Thirdly.     When  n  z=z  —  \ 

(279)  becomes  infinite.  But  the  infinite  form  may  be  avoided 
by  means  of  an  infinite  arbitrary  constant,  such  as  that  of 
(2S0).  In  this  case,  (2S0)  assumes  an  indeterminate  form, 
the  true  vahie  of  whicli  may  be  ascertained  by  means  of  B. 
II.  §  104.  For  tlie  differentiation  of  the  terms  of  (280)  rela- 
tively to  n  gives,  by  (Vol.  I.  481), 


/ 


,^  ar7,.„.(x-+i  — :r"J-i) 


Xq*  (/,.„.  (/i  +  l) 

=  a  (x«+i  log.  X  —  x^-^^  log.  Xq) 
=  a  (log.  X  —  log.  xo),  (281) 

or  omitting  the  arbitrary  constant  a  log.  x^ 


/ 


a 


=  a  log.  X.  (282) 


21.  Corollary.  Every  algebraic  polynomial^  being 
the  sum  of  jiionomials  of  the  form  ('^74),  may  be  inte- 
grated by  integrati7ig  its  terms  separately ;  and  a?iy 
function  can  also  be  integrated  by  this  process ^  which 
can  be  reduced  to  such  a  polynomial. 

22.     Examples. 

13  1 

1.  Integrate     6x5+x-+5V^-2 ^  _{- 8  x-9. 

o  i  '  x^ 

Ans.     x6  +  f  x^  +  3  x^  +  J-  x-4  —  x-8. 

2.  Integrate     SV^x  +  Jx-J.  Ans.     2xi-\-\/x. 


<5>  25.]  RATIONAL    FUNCTIONS.  71 


Jntegration  of  rational  fractions. 


8.  Integrate     a  x^  -{- b  x^  -\-  -  -{-  h. 

Ans.     ^ax'^-{-^bx^-\-c  log.  x  -[-  hx. 

4.  Integrate     x~^  {x^  -\-  x  -{-  1)2. 

A71S.     ^x3-|-x2-|-3z4-21og.  2  — ar-i. 

5.  Integrate     2;(x  +  x-i)2.         v4?i5.  :^  x^  +  2;2 -|- log.  a;. 

23.  Corollary.  The  substitution  of  y.  x  for  x  in  (279)  and 
(282),  gives  by  §  18, 

a  (w    x')"+^ 
/.«(c;.x)"^..<p.x^      IVi      ^  ^^^^^ 

/.  fiiil^j^  ^  a  log.  9 .  X.  (284) 

24.  Corollary.     Let 

<^.x  nz  6  X -j- c,     d^.cp.xzu^b,  (285) 

(283  and  284)  become,  by  dividing  by  6, 

25.  Pi^uhlem.   To  integrate  a  rational  fraction. 

Solution.  Let  the  fraction  be  reduced,  as  in  B.  IV.  §  16,  to 
a  mixed  quantity,  of  which  one  part  is  an  integral  polynomial, 
and  the  other  is  a  rational  fraction,  in  which  the  degree  of  the 


72  INTEGRAL    CALCULUS.  [b.  V.  CH.  II. 

Integration  of  rational  fractions. 

numerator  is  less  than  that  of  the  denominator.    If  this  second 
part  is  denoted  by  f.  x,  (*21G)  and  B.  IV.  '^  15  give 

The  integral  of  (288),  relatively  to  x,  is  by  (287)  and  §  11 

and  (289),  added  to  the  integral  of  the  polynomial,  is  the  re- 
quired integral. 

26.  Corollary.  Since  (288)  is,  by  the  process  of  B.  IV. 
^  4,  Ex.  9,  reduced  to  the  sum  of  several  fractions  of  the 
form 

w=^r  ^'''^ 

(289)  is  the  sum  of  their  integrals,  or  is  itself  by  (286  and 
287)  the  sum  of  several  terms  of  the  form 

f'x 

when  n  is  greater  than  unity,  and  of  the  form 

/.a:Jog.  (:.  — xj  (292) 

when  n  is  unity. 

27.  Corollary.  When  the  given  rational  fraction  is  a  real 
function,  it  follows  from  B.  III.  <5§  37 -o9,  that  x'^,  the  con- 
jugate of  Xq,  furnishes  a  fraction,  corresponding  to  (290) 

f.  X  ' 
T^ — V  J  (293) 


-*•• 


<^  27.]  RATIONAL    FUNCTIONS.  73 

Integration  of  rational  fractions. 

such  that  if  fJ  Xq=  A-]-B\/—l,  (294) 

then  /.'xq'  =  A  —  Ba/~1;  (295) 

and  if  we  put       (z  —  Xq)""-^  —  ^+  Y \^  —  1,  (29G) 

in  which  X  and  Y  are  functions  of  z,  we  have 

(x  —  x^y-^  =  X—  Y^/  —  1.  (297) 

Hence  the  sum  of  (291)  and  the  conjugate  fraction  is,  sup- 
posing 

x^  =  a  +  bA/—l,  (298) 

1       {A+Bs/-\)(X-Y^/-\)  +  (A-Bx/-l){X+Y^/-l) 
n—l '  {x  —  Xq)^-^    (x  —  V)"""^  ■  " 


1  2AX+2B  Y 


(299) 


n  —  i    (x2  — 2ax  +  a2-(-62)"-i' 
which  is  a  real  function. 
In  the  same  way,  since  (95)  gives 
log.  {x^x^)—  log.  (x  —  a  —  6  V  —  1) 

=  iJog[(^— «)'+2'']-tan.[-i]  — . V-1,  (300) 
the  sum  of  (292)  and  its  conjugate  is 

A  log.  [(x—  a)2  +  62]  +  2  i?  tan.[-i]  -^  (301) 

which  is  real ;  so  that  the   required  integral  is  thus  entirely 
freed  from  imaginary  quantities. 


% 


74  INTEGRAL    CALCULUS.  [b.  V.  CH.  II. 


Integration  of  rational  fractions. 


28.     Examples. 
1.  Integrate 

x7  _|_  1 0  j6  4.  36  a:5  _|_  67  a;4  _)_  68  x 3  _|_  29  2-2  __  4  2;  __  7 
~  (x2  +  2x-j-2;2  (x4-  1)3  (X—  1)  ■ 

Solution.  First.  When,  for  this  example, 

^0  =  1, 
we  have  w  =  1,    f.x^z=\, 

and  (292)  becomes  log.  {x  —  1). 

Secondly.     When  ^0  =  —  ^> 

we  have  for  w  =  3,    f'.x^zzil^ 

so  that  (291)  becomes  —  -— — -— —  : 

2(z-{-  1)^ 

and  we  have  for  w  z=  2,       f.x^z^z  0, 

for  W  =:  1,        f.  X(j  =z  0. 

TJdrdhj.     When         a;o=r  —  1+V— 1, 
we  have  for  n  =r  2,       f.xQ  =  —  ^  ; 

so  that  by  (294  and  295) 

^  =  —  J,         5  =  0, 
X—x-\-\,      Fzn  — 1, 

and  (299)  becomes  ^^^-^_hi__ 

We  have  also  for        n  :=  1,     /,*  x^  z=  ^, 
so  that  by  (294  and  296) 


<5,  28.]  RATIONAL    FUNCTIONS.  75 

Integration  of  rational  fractions. 

A=0,      B^—l, 

and  (301)  becomes  —  ;^  cot.[-i](x  +  1). 
The  required  integral  is,  therefore, 

log.(x-l)_i(.+l)-+^^^5^jdy__jcot.[-.l(.  +  l). 

X5   J-  1 

2.  Integrate 


(x_l)-^(.T  +  2) 


_      _  X 1 

3.  Integrate 


4.  Integrate 


(x+l)(x_2)- 
^ns.     f  log.  (X  +  1)  +  ^  log.  {x  —  2). 


(x+  1)  (x  — 2)2 


^n5.     Jx2+-A__2iog.(x-2)  +  |log.(x  +  l). 

^    _  n  X  -4-  m 

5.  Integrate    — — ^-rn- 

n 

Ans.     -  log.  (x2  —  2  a  X  -(-  «^  +  &^) 

+  !iii  +  i:!tan.t-']?^.  (302) 

_    _  71  X  -4-  m 

6.  Integrate    ^^^^. 

^„,.     "log  /x5+*\_-_;!L^tan.[-i]^^.      (303) 


76  INTEGRAL    CALCULUS.  [b.  V.  CH.  II. 

Integration  of  rational  fractions. 

7.  Integrate 


a;2  +  1* 


Ans.     tan.Ml  x  z=  cot.M]  -.        (304) 

X 


n  X  -\-  m 
8.  Integrate 


{n  a-\-ni)  X  —  n  {n^  -\-^^^)  —  ^^  (^ 


Ans. 


2^2(2:2— 2ax+a2^62) 

^  na-{-m         .   n      ^  /on-x 

-  A -! tan.[-i] .  (305) 


<5>  30.]  IRRATIONAL    FUNCTIONS.  77 


Integration  of  irrational  functions. 


CHAPTER   III. 

INTEGRATION    OF    IRRATIONAL    FUNCTIONS. 

n 

29.   To  integrate    f.[x^j^{ax-{-  6)]. 

Solution,     Let  y  =  ^  {ax-\-h)y  (306) 

whence  x  =  ^^ ,  4y  ^  =  -^^ — >  (307) 

and  by  §  19 

=  ,.,,(£=5,,). -p.      ,308, 
30.     Examples. 

n 

1.  Integrate    \/{ax-\-hY'. 

Solution.     Equation  (308)  becomes,  in  this  case, 

•^        ^         '     '       «/  a  {in-\-n)a 

n\/{nx  -\-  Z>)™+»] 
(w  -j-  w)  a 

_    _  \/^  -f-  1 

3.  Intetrrate    — -. . 

°  S^x—  1 

^7i5.     X -(- 4\/x  +  41og.  (\/x  —  1). 

7* 


78  INTEGRAL    CALCULUS.  [b.  V.  CH.  III. 

Integration  of  irrational  functions. 

\/2;4-  1 

3.  Integrate    -^ — . 

a/x  —  1 

6 

Solution.     Let  y  i=z  \/  % 

and  (308)  becomes 

n 

4.  Integrate     x  s/  (x  -\-  a)  -\-  \/  [x  -\-  a). 

5.  Integrate     ^^  ^ ,  ,  ^-    T,    ^  ^ 

^  \/  {ax  -\-  b) 


Ans.     T — j — — . 

a{?i  -\-  i) 


31.  Problem.   To  integrate  f.  [.v,  ^(ax-  -\-  bx  -\-  c)]. 

5  52 4  rt  c 

Solution.     Let  x  =^  y  —  --,  m  =  — - — - — ,      (309) 

2a  4  «2  ^       ' 

1)2  —  A.  a  c 
whence     g  x^4- 6  a;  +  c  =  «  ?/2 

—  a{y^-^m)  (310) 

c/,.yx=l;  (311) 


^  31.]  IRRATIONAL  FUNCTIONS.  79 


Integration  of  irrational  functions. 


and  by  §  19, 

f.f.[z,  ^/{ax^+bxJrc)]=ff.^j-^^,^/[a{,f-m)]']  (312) 

There  are,  then,  several  cases : 

First.  When  a  is  negative  and  m  negative,  the  radical 

s/[a{y2  —  m)]  (313) 

is  always  imaginary,  and  the  integral,  being  imaginary,  admits 
of  no  real  solution,  and  may  be  solved  as  in  either  of  the  other 
cases  which,  in  this  case,  become  imaginary. 

Secondly.    When  a  is  positive  and  m  negative,  y^   must  be 
greater  than  m,  when  (313)  is  real,  let,  then, 

z=.s/{y^  —  m)—y,  (314) 

whence        (y -\- zY  ~y^ -\-^y  z -\-z^  =  y^ —m         (315) 

y  =  -^-^z  .        (316) 


2z 

2z2 


^-3/ =  .7^ -J  (317) 


and  (312)  becomes 
v^         Thirdly.     When  ni  is  positive,  let 

m 

,2  ^  -JL-^,  (320) 


80  INTEGRAL    CALCULUS.  [b.  V.  CH.  III. 

Integration  of  irrational  functions. 

\////.(22-|_rt) 

whence  y  =: ^ —  Ky^*-) 


«  —  z^ 

4 

as/  m .  2 

(a-;:2)2 

2 

«\/  m.  2 

V[«(y^-'»)]  =:  -r^-  (323) 

and  (312)  becomes 

^      r\/ffl.(22-|-Qr) h    2a\/m.zn   \as/m,z 


32.     Examples. 

1.  Integrate    — — Trs—, ; • 

*  V(«3;2  +  ^z+c) 

Solution.     In  this  case  (312)  becomes 

J'  ^[a{y2—n^  (^^^) 

(319)  becomes,  by  reduction, 

—  f-  —, = -f-  log-  ^ 

«/     \/a.z  \/a 

= —  ;^- log- [V(y^— ^)  — y]. 

or  since 

—  m 


^{y2  —  m)—y  = 


\^{y-—m)-{-y 
log.  s/[y^  —  ni)—y  =  log.  (— ?w)  —  log.[\/(y^  — "0+^] 


<5>  32.]  IRRATIONAL    FUNCTIONS.  81 


Integration  of  irrational  functions. 


in  which >- ^og-  — "*  ^^^^y  be  omitted,  as  in  art.  3. 

Again  (324)  becomes 

which,  when  a  is  positive,  is 

_!_  ]  cr  ^+^^  —  J_  io(T  V(y+V^)+V(y— ^^^) 
=  _!_., og.[V(.^-»)+.]-3-^       W 

but  when  a  is  negative  (327)  is  as  in  (803) 

is/— a  z  \/—a  ^  Ww*+3// 

The  form  of  this  last  solution  may  be  changed  in  several  ways, 
which  will  often  be  useful ;  thus,  let 

,..tan.[-]l(^^^^^\  (330) 

\i  s/  m  —  y\ 

whence      tan.  ^  z=z  \\  —, ■ —  I 

-^Xs/m  -\-  yf 

2  \/  m 
sec.2  5  =.  1  +  tan.2  6  — 


COS.''^  A  = 


\^m  -f-  y 

A^m  —  y 


sin. 2  5  z=z  tan.2  a  .  cos.2  a  = 


2x/wt 


m-7/2       4r/(r/x2-|_6x-|_c) 

sin. 2  2^=4  sin. 2  a .  cos.2  ^  = =  — — 

m  ^ac  —  0^ 

y  b  ■4-2  ax 

COS.  2  d  =  2  cos.2  6^1  —  -^—  — 


Vwt""  V(^--4ac) 


82  INTEGRAL    CALCULUS.  [b.  V.  CH.  III. 

Integration  of  irrational  functions. 
and  (329)  gives 

J'  /s/(ax^+bx-\-c)~ \/—a  ^        ^ 

=  —, sin.t— ^J — -— ! ~ — •' 

V — a  v(o    — 4^6;) 

1  h  +  2ax 

=. cos.L— ■'J ■ 

1         .     r    n    ::pb^2ax 


2.  Integrate    - -.  Ans.     sin.[-i]  z.         (332) 

3.  Integrate    ^^/_^^,^.  (333) 

Ans.     log.  [x  +  V  ( 1+  a;^)]- 

4.  Integrate    ^^J_^y      .  (334) 

^ns.     log.  [x  +  V  (^^  —  5  )]• 

5.  Integrate    ;;7(X^,^. 

Ans.     —  (^x2+ 1)^(1  _x2). 

.  _1_  1         V(^+6  2:2)— z^6— ^q 

-    1     1^,,  V(«+^^")-V« 


2/s/«      "^    V(«  +  ^^^)+V« 


_       1        sin.[-]ij-^. 
a/ —  a  X  ^       0 


"      V 


<§>  33.]  IRRATTO  ^M,    FUNCTIONS.  83 

Integration  of  irrational  functions. 

^        T  1 

7.  Integrate 


Ans.     log. 


8.  Integrate    — —- r-. 


Ans.      log. 


^{l  +  z^)- 

(336) 
-1 

X 

(337) 
—  1 

X 


^  '"''^'''^  Wi^)  ^'''^ 


Ans.     — sin.t— 1]  -  or  sec.C— ^^x. 

X 


10.  Integrate 


^(x2+xy 

Ans.     \og.[i  +  x+^{x^  +  x)], 

n 

33.  Problem.  To  integrate  f.fx,    I.^L+Af.1  (339) 

L-      '^J   c  ~|~  it  X  —^ 


Solution.     Let 


a-\-h  X 
2'"=7TI-x'  (340) 


whence  x  — 


hif—b' 


(341) 


,  nih  c  —  ah)  ?/"— 1 

^'■'■^=    \uy^Jk)~'  (342) 

and,  by  §  19,  the  integral  of  (339)  becomes 

r    f  V"!—^     ,/"!      ""  (^  C—ah)j/n-l 


84  INTEGRAL    CALCULUS.  [b.  V.  CH.  III. 

Integration  of  irrational  functions. 

34.    Examples. 
,.  Integrate      jl^. 

Ans.    V(l-^)V{l+^)--Iog.[V(l-^)  +  V{l+^)] 
- 1 V  3  ta„.[->]     3    V^.Vd-^ _ 


2V{i+^)-V(i-»:) 

1  3      1     y. 

2.  Integrate    — — ; — —  .    I- . 

3   /I— a:\* 

T 


^-  ^j(iT-D 


35.  Problem.     To  integrate 

^™"V- [/>/(« +  ^^")>  ^^  ^j"],  (344) 
when  m  is  exactly  divisible  by  n. 

Solution.     Let                «  +  6  x"  r=  y' ;  (345) 

whence                            a:"  =  ^:— - — ,  (346) 

.»  =  (?l^)==".  (347) 

The  differential  coefficient  of  the  logarithm  of  (346),  gives 

~^^^—=.y—,  (348) 
whence 

X— irf      z  -  ^y^"'  ^y'-«\  n  /349X 


<5>  37.]  IRRATIONAL    FUNCTIONS.  85 

Integration  of  irrational  functions, 
and,  by  §  19,  the  integral  of  (344)  becomes 

m  m 

36.   Corollary.  When  q  z=z  2  and  2  m  is  divisible  by 
71,  (350)  is  integrable  by  ^  31. 

37.     Examples. 

5 

1.  Integrate      x^  ^{a  -\-  b  z^). 
Solution.     In  this  case 

m  =i  4,     7i  z=  4,     q  z=z  5  ; 

5 

whence  ?/ z=:  ^(« -|-6  x*) 

5V(«+ ^2:^)6 


24  6 


5 

2.  Integrate     a:^  ^^^j  _|_  j  3-2). 


3.  Integrate 


8 


86  INTEGRAL    CALCULUS.  [b.  V.  CH.  III. 

Integration  of  irrational  functions. 


a;2 
4.  Integrate    — — — . 

Ans.     —  J  X  V(l  —  x2)  +  J  sin.M]  x, 

38.  Problem.     To  integrate 

x'n-i  {a-\-h  x'^)!  f.  (z")  (351) 

when \--2san  nte^er. 

n    ^  q  ° 

Solution.     Let             ax-^  -^  b  z=z  y%  (352) 

whence  x"  =  — -,  (353) 

771 

The  differential  coefficient  of  the  logarithm  of  (353)  gives 

X        ■"  y'i  —  6' 

whence 


(356) 


'=-y  \2f  —  b/     n(y'i  —  bY  ^        ' 

and,  by  §  19,  the  integral  of  (351)  becomes 

m        p 


§  40.]  IRRATIONAL    FUNCTIONS.  8T 


Integration  of  binomial  irrational  functions. 


1.  Integrate 


39.     Examples. 
1 


Ans.    -(-1 _3^WVM-_if!)!\ 

5 

2.  Integrate    ^  ■    ~ L., 


8ax8 


40.  Problem.     To  integrate 

z'"  (a  +  6  x")^  (359) 

WjAe?i  m  and  n  are  positive  integers^  and  p  is  a  posi- 
tive fraction. 

Solution.     First.     Let            v  z=z  x\  (360) 

whence                               d,,.v  z^  sx^—'^,  (361) 

in  which  s  is  to  be  taken  of  such  a  value  as  may  be  found 
most  useful  \  let,  then, 

ud,.v  =  x^  {a-\-h  x^)P  ;  (362) 

whence               u  =^  -  2"i-s+i  (^^  _[_  j  ^r^y^  (363) 


d,.U==r 


m-s-\-l 


{a+h X") P  +  ^^ x^-'+^{a+h  x")?-! ;  (3G4) 


8S  INTEGRAL    CALCULUS.  [b.  V.  CH.  III. 

Integration  of  binomial  irrational  functions, 
or,  since      {a-{-bx''y=z  {a  +  bx")  (a-{-bx")P-\         (365) 
(7}i-s-\-l)a-\-(7n-s-\-l-\-n  p)hx"  ,     ,  ,    „,     .    ,con\ 

and,  by  (262),  the  integral  of  (359)  is 

-z^'+Wa  4-  bx'')P 
s 

and,  if  s  is  taken  such  that 

m  —  s  +  l+wp  —  0,  that  is,  s  z:^  m -\- \ -\- n  p,     (368) 
(367)  becomes 

a;'"+i(rt  +  6x")^  +  awp/.  x'"  (a  +  6  x")p-_i) 
wt  -J-  1  -|-  w  p 

The  value  of  the  required  integral  is  thus  made  to  de- 
pend upon  that  of  an  integral,  in  which  the  exponent  of 
the  binomial  (a  +  5.^"')  is  diminished  b^  unity;  the 
value  of  this  new  integral  may,  by  the  same  formula 
(369),  be  made  to  depend  upon  that  of  an  integral,  in 
which  the  exponent  of  the  binomial  is  still  farther  di- 
minished ;  and  so  on  until  the  exponent  of  the  binomial 
is  reduced  to  a  fraction  less  than  unity. 

Secondly.     Instead  of  (360)  let,  now, 

v  —  {a  +  b  x^^y ;  (370) 

in  which  s  is  to  be  taken  of  any  value,  which  may  be  found 
useful ;  whence 

d,.v=znbs  x"-i  {a  +  b x^-^,  (371 ) 


<J  40.]  IRRATIONAL    FUNCTIONS.  89 


Integration  of  binomial  irrational  functions, 
and  if  u  is  taken  so  as  to  satisfy  (3G2), 

u  =  -4-  .  x"»-«+i  (a  4-1)  z")P-'+^  (372) 

71  b  S  V         1  / 

±,u=z  -11^  x^^-n(a+h  xy-'+^  -i-Pz!+l  x'^ia  +  b  x^)p-' 
nb  s  \     I        /  '         s 

(a(m-n-\-l)  ,  j)i-{-l-[-7ip-ns      \,     ,  ,     ,         ,r».^r»v 

71  b  s  s  ' 

and  if  s  is  taken  such  that 

77Z  -|-  1  +  77J5  —  n  s  z=  0,  that  is,  s  z=.  +  P      (^^'^) 

fv 

(373)  becomes 

«f%Mzr  -i — — — -^ — ^  x^-^  («  +  5  x'')P-^  (3/d) 

6(7?f-|-l-|-72  2j) 

and,  by  (262),  the  integral  of  (359)  is 

2"'-"+i(«+5  x")p+i  —  a{m  —  n-{- 1 )/.  x™-"(a  +  &  x")^ 

6(m  +  1  +wp)  '  ^'      ' 

ill  which  the  exponent  of  the  factor  x"^  of  the  binomial 
is  diminished  by  that  of  x^  in  the  binomial,  and  tliis 
exponent  tnaij  by  a  repeated  application  of  (376)  be  still 
farther  diminished  until  it  is  less  than  7i. 

Tliirdly.  By  the  successive  use  of  (369  and  376), 
the  required  integral  may  be  made  to  depend  upon  one 
of  a  similar  form^  in  which  the  exponent  of  the  binomi- 
al {a  -\-  b  x"")  is  less  than  unity ^  and  that  of  its  factor 
is  less  titan  n. 

8* 


90  INTEGRAL    CALCULUS.  [b.  V.   CH.  III. 


Inteffration  of  binomial  irrational  functions. 


Fourthly.  The  development  of  (a-\-  h  x"Y  may  be  effect- 
ed by  the  binomial  theorem,  either  according  to  ascending  or 
descending  powers  of  2 ;  it  being  better  to  use  the  ascending 
powers  when 

X"  <  |,  (377) 

and  the  descending  powers  when 

^  >  I-  (378) 

In  one  case  the  integral  of  (359)  is 

qp  2;7n+i       paP-^  b  z"H-"+i       p{p-l  )aP-^ b^ x™+2»+i 
—  m-\-i^    m  +  71  4n~"  +   l.2.(m-\-2n-\-l)    "^     ^* 

[1  p  bx" 

/^f-j-l       ?;i-|-7i-f-i     a 

^1.2.(m  +  2w  +  l)  \   a   /    ^         J  ^       ^ 

and  in  the  other  case  • 

_  r  ^  a-^_^L^ ! I-  &c.  (380) 

=  fe?  x'lP+'W+l  i I .  -; h  &C.  I 


<§)  41.]  IRRATIONAL  FUNCTIONS.  91 

Integration  of  binomial  irrational  functions. 


41.     Examples. 

1.  Reduce  the  integral  of  x*  {a-{-bx^)^  to  depend  upon 
one,  in  which  the  exponent  of  a  -\-  b  z^  is  less  than  unity,  and 
the  exponent  of  its  factor  is  less  than  3. 


Solution.     By  putting  in  (369) 


m  =4,     717=3,    i?  =  J, 
it  gives 

and  by  putting  in  (376) 

m  =  i,     7i=z3,    p  =  i, 
it  gives 

SO  that,  by  substitution, 
f.xHa  +  bx'r={^^^x^-\--°-.—ya  +  bx^f 

2.  Develop  the  integral  of  x  (a-|"  ^  •'^")"  according  to  pow- 
ers of  X. 

Solution.     By  putting  in  (379  and  380) 

m  =  1,     71  =1  3,    p  =  i, 


92  INTEGRAL    CALCULUS.  [b.  V.  CH.  III. 

Integration  of  binomial  irrational  functions. 

=V(6x^)[f+^3-Tv(-^)^-&c.] 

4 

3.  Reduce  the  integral  of  X*  (a2  —  3-2^3  ^q  depend  upon 
one,  in  which  the  exponent  of  a~  —  x^  is  less  than  unity,  and 
that  of  its  factor  is  less  than  2. 


Ans.  {^\  x5  —  ^2^\  a2  x^  —  f^  a^  x)  (a^  —  x^)~^ 

1 

4.  Develop  the  integral  of  («2 — x^)^  according  to  powers 

of  X. 

2     /  a;2  2;*  \ 

Ans.    a'2\\-^l.  —  —  -^\.  —  +&L0.^ 

2. 

5.  Develop  the  integral  of  x(l  -^x^Y   according  to  powers 

of  X. 

Ans.     z2  (^_|-_2_2.3.__i^a^6_|_&c.) 

or  x>  (^  +  2a;-3  4-_i_.x-6+&c.) 

42.  Problem.   To  integrate  (359)  for  all  real  values 
of  771,  n,  and  p. 

Solution.     First.  When  m  is  a  negative  integer  and  n  a 
positive  integer,  the  substituting  of  7n-\-n  for  m  in  (376), 


<5»  41.]  IRRATIONAL    FUNCTIONS.  93 


Integration  of  binomial  irrational  functions. 

freeing  from  fractions,  dividing  by  a  (m  +  1),  and  transposing, 
give  for  the  required  integral 

X  "»+^  (a+b  X")  P+^—b  (m  +  l+Jip+?i)  f.  x^+^  {a+b  xy  .^^.. 
a(m+l) ^ '(^^^ ) 

which  formulaj  since  ni  is  negative,  serves  to  increase 
the  exponent  of  the  factor  of  the  binomial  under  the 
sign  of  integration,  until  it  becomes  positive  but  less 
than  n. 

Secondly.  When  p  is  negative,  the  substitution  of  p  -|~  1 
for  p  in  (369),  gives  by  reduction  for  the  value  of  the  requir- 
ed integral 

which  formula  serves  to  increase  the  exponent  of  the 
binomial  under  the  sign  of  integration,  until  it  becomes 
positive  hut  less  than  unity. 

Thirdly.  When  m  and  n  are  fractions  and  n  positive,  let 
the  common  denominator  of  m  and  n  be  /,  and  let 


x  =  y'  (383) 


whence  d,yX=zly^-^  (384) 

and,  by  §  19,  the  intregal  of  (359)  becomes 

/  l7f'^+^-^a  +  b  y^^^Y  (385) 

in  which  the  exponents  of?/  are  integers,  so  that  it  may- 
be integrated  by  ^  40,  or  the  preceding  part  of  this  sec- 
tion. 


94  INTEGRAL    CALCULUS.  [b.  V.  CH.  III. 


Integration  of  binomial  irrational  functions. 


Fourthly.  When  n  is  negative,  a  simple  algebraic  reduction 
gives 

x^  [a  +  h  x")^  =  z'»+"P(6  +  az-")^  (386) 

the  integration  of  which  may  be  efTected  by  «§)  40  or 
the  preceding  part  of  this  section. 

43.     Examples. 

i_ 

1.  Reduce  the  integral  of  z  — 2  (i_|_ a; 3 )-;i  to  depend  upon 

one,  in  which  the  exponent  of  the  binomial  is  positive  and  less 
than  unity,  and  that  of  its  factor  is  positive  and  less  than  3. 

Solution.     The  substitution  of 

771  =  —  2,    ?l=r:3,   2?=:  —  ^,    a=:\,    6=1, 

in  (3S1),  gives 

/2-2  (1  4.2;3)-^  =  — 2-1  (l+a:3)^-|-y:2(l  +  x3)-^, 
the  substitution  of 

77^  zz:  1 ,    w  z=  3,  jP  =  —  4-,    a=i  1,   b  =  l, 
in  (382),  gives 

/  X  (1  +  23)--^=—  ia;2  (1  -j-a;3)f  _|_oy:2;  (i  -f  x^)?"- 
Hence 

/  2-2  (1  -|- z3)-^___  (a;-l_|_  ^  a;  2)  (1  _|_  a^3)f  _[_2y:a;  (l_j.2;3)l 

_2 

2.  Reduce  the  integral  of  2-2(1  -^x^)  ^  to  depend  upon 
one,  in  which  the  exponent  of  the  binomial  is  positive  and  less 
than  unity,  and  that  of  its  factor  is  positive  and  less  than  3. 

Ans.     I  (2  —  2-2)  (l+xs)^—/  (l  +  23)i 


§  44.]  IRRATIONAL  FUNCTIONS.  95 


Integration  of  binomial  irrational  functions. 


L2 

3.  Reduce   the    integral   of    {l-\-x^)    ^    to   depend  upon 

one,  in  which  the  exponent  of  the  binomial  is  positive  and  less 

than  unity. 


-^    -    ■  I 


Ans.     J  X  (I  +x^)  ^  (3  +  a;5_2;io)  +  ty:  {I  +  x^)' 

2  4 

4.  Reduce  the  integral  of  x-^  {a  -{-bx'^y  to  depend  upon 
one  of  the  same  form,  but  in  which  the  exponents  are  integral, 
except  that  of  the  binomial. 

Solution.  In  (383)  we  have,  for  this  case, 

so  that  (385)  gives 

/  ^  {a-\-hx^Y  —  15/ ?/24  (^a  +  h  y^^y. 

4  2 

5.  Reduce  the  integral  of  x^  (a  -\-  h  x^Y  ^^  depend  upon 
one  of  the  same  form,  but  in  which  the  exponents  are  integral, 
except  that  of  the  biomial. 

Ans.     I5f.  y26  (^a  +  b  i/^^y^ 

,3.  3 

6.  Reduce  the  integral  of  x'^  (a -\- b  2;  ~2)^  to  depend  up- 
on one  in  which  the  exponent  of  x  in  the  binomial  is  positive. 

3 
A71S.    J.  {b  -{-  a  x^)^. 

44.  Problem.  To  find  the  value  of  the  definite  inte- 
gral 

fl  x^  {a  +  b  xy  (387) 

in  which  c  =  ^--^  (388) 

0 

and  nij  n^  and  p  are  positive. 


96  INTEGRAL    CALCULUS.  [b.  V.  CH.  III. 

Value  of  binomial  definite  integral. 
Soliti.n.     Tiie  substitution  of 

a  =  — -  6  c"  (389) 

reduces  (387)  to 

^''/o  ^"^  (^'' —  c"Y  =  {— by  f^  x"^  (t"  — z")?.      (390) 

First.  The  term  of  (3G9) 

a;»«+i  {a-\-h  x^'Y  —  JjP  i"^+i  (x'*  —  c^'Y  (391 ) 

is  zerOj^wheii         z  =  0,    and  when   x  z=z  c.  (392) 

Hence  (369)  gives  for  the  value  of  (387) 

""''P      -.Sl,x-{a-\-bxy-\  (393) 


m-\-\-\-np 


and,  in  the  same  way,  by  changing  p  top  —  1,|9  —  2,  &c., 
(309)  gives 

&c. 

The  substitution  of  each  successive  value,  in  the  preceding 
one,  gives  for  the  value  of  (387),  if  pQ  is  the  greatest  integer 
in  p, 

{^  7l)P0p{p—\)  (p— 2)   .  .  .  .  (  p— po  +  1  ) 

(m-{-l+np)[m-}-l  +  n{p—l)]....[m+i-{-n{p—pQ-\-\)] 
X  /o  x'"  (a  +  6  x")2'-'o  (396) 


<5>  45.]  IRRATIONAL    FUNCTIONS.  97 


Value  of  binomial  definite  integral. 


Secondly.    In  the  same  way,  (376)  gives 
/„%  z'-ia  +  bx-y^-  4^":p^]/o-  ^•"-" (°+6^")'  (397) 

and  for  the  final  value  of  (387),  if  h  is  the  greatest  integral 
number  of  times,  which  n  is  contained  in  tw, 

Thirdly.  The  series  (379)  gives  for  the  value  of  (387) 

45.   Corollary.    In  the  particular  case,  in  which 

m=:0,      71  =  2,     p=:  — J,  (401) 


we  have 

n.  —  —  h  r.^ .      r  ■ —  a/  — 

b 

and  by  (331) 


a  =  —  6  c2,     c  =  V  —  X'  (^^^) 


/V(^^^  =  ~V=6^''-^"'V(^     ('^') 


whence 


,7  0  v(«+ox2)  ^ — 5  '  y' J 


1  TT 

(404) 


9 


98  INTEGRAL    CALCULUS.  [b.  V.  CH.  III. 

Value  of  binomial  definite  integral. 


46.  Corollary.    It  follows  from  (404),  that  if 

b  =  —g2,     cz=  ^  (405) 

s 


which  c  = . 

g 


47.    Examples. 

oV(«— ^2a;2)' 


— 77 -^r-K7,  in 


Solution.     The  substitution  of 

77«  zzr  4,      W  =:  2,     p  z=z  —  J,      A  =r  2 
in  (399)  gives,  by  (406), 

y'c  %^  a2   S.l    /*c         1  _3a2     TT 

0  V(«— ^' ^•')  ~  ^  *  T:^JoV(a-g'  x2)  -  8^  •  ■2"- 

0    V(«^ a;2) 

1.3.5      TrflS 

^"^-     274-6-2- 

.      .,     ^ -• 

0 V(«   —^  ) 
Solution.     Equation  (399)  gives 

^«  a;3  2a2    r»a  a: 

JoV(«^  — ^^"~~3-./oV(a2  — x^* 


<5>  47.]  IRRATIONAL    FUNCTIONS.  99 

Value  of  binomial  definite  integral. 
But  by  (324) 

whence  /  .    ..   ^ :rr  =  a» 


and  f  ,—r, rr  ==  f  a^- 


p.  2;3 


2.4 

^W5.     — — -  a^. 
o  .  5 

5.  Find  the  value  of  the  definite  integral  J^. \/  {a  -\-  h  x^) 

where  c  ■=.  s/  —  -. 
o 

Solution.     The  substitution  of 

m  =  0,       n  —  2,      p  =  iy 

in  (393)  gives  by  (404) 

/iC  fj       /*C  1  TT  /7 

Q  vv  -r       ;       2*^  0  \/{a+bx2)      ^A^—b      ^       ' 

6.  Find  the  value  of  the  definite  integral  /*.  \/(a2  — x^), 

Ans.     — - — . 


7.  Find  the  value  of  the  definite  integral  f^.  i^s^iofi  —  x2). 

Ans,     I .  -J-. 

8.  Find  the  value  of  the  definite  integral  J^.  %  V  («^  —  a^^)- 

Ans.     ^a^. 


100  INTEGRAL    CALCULUS.  [b.  V.  CH.  III. 

Value  of  binomial  definite  integral. 


9.  Find  the  value  of  the  definite  integral  f^.  x^\/[a^  —  x^). 

Ans.     I .  ^  a^. 

5 

10.  Find  the  value  of  the  definite  integral  f^.  (a^  —  x^)^. 

Ans.     -^^  n  a^. 

11.  Find  the  value  of  the  definite  integral  f^.x^^a^ — x^)^. 

Ans.     ■g-.fV^^^' 

12.  Find  the  value  of  the  definite  integral  f^,x^{a^ — x^y. 

Ans.     f  .  -i  a"^. 

13.  Find  the  value  of  the  definite  integral  /*.  x(a^  —  x^)'^. 

A?is.     -f  a^. 


O^.        (^>c-T^ 


<J  48.]  LOGARITHMIC    FUNCTIONS.  101 

Integration  of  logarithmic  functions. 


CHAPTER   IV. 

INTEGRATION    OF    LOGARITHMIC   FUNCTIONS. 

48.  Problem.   To  integrate 

f.  X.  (log.  F.  xf  (408) 

Solution.  First.  When  n  is  positive,  the  substitution  of 

u  =  (log.  F.  xY,  d,,  v—f.x,  (409) 

in  (262)  gives  for  the  integral  of  (408) 

(log.  F.  .ff.f.  ._„/(i^.^lfr_L^^iW:/_^(4io) 

by  which  formula  the  exponent  of  the  logarithm  is  di- 
minished by  unity,  and  may  be  still  farther  reduced  by 
the  repeated  application  of  the  same  formula. 

Secondly.  When  n  is  negative  and  differs  from  —  1,  the 
substitution  of 

.=  (Iog.F.x)..+S«=^^J^,         (411) 
in  (262)  gives  for  the  integral  of  (408) 

by  which  the  exponent  of  the  logarithm  is  increased  by 
unity. 

9* 


102  INTEGRAL    CALCULUS.  [b.  V.  CH.  IV. 


Integration  of  logarithmic  functions. 


Thirdly.  When  n  —  —  l  (413) 

the  only  useful  reduction  occurs  in  the  case  of 

/.  X  =  ^^  ^  rf,  log.  F.  X  (414) 

in  this  case,  since         d^.  log.^  F.  x  =^  — — - —  ,     (415) 

the  required  integral  is    f.  — — -, — ^=^—  —  \og.^' F.  x.  (416) 
^  ^  J    F,  X  log.  F.  X 

Fourthly.  The  particular  case  of  (414),  gives  also  a  dif- 
ferent solution  of  the  general  problem;  for  in  this  case  the 
integral  of  (408)  is 

f.  (log.  F.  xy.d\  log.  F.x—  ^    ^'^^  _^ .     (417) 

In  other  cases,  the  integration  can  only  be  advanced 
in  the  form  of  a  series. 


49.     Examples. 
1.  Integrate         x"^  (log.  x)2. 

Solution.  First.  When  m  differs  from  —  1,  in  which  case 
(410)  gives 

/x'^+Hlo^.  a;)  2  2         /*    „  , 

x-  (log.  X.  2  = 5l^— i — _    /.  x^  log.  X 

X'"  (log.  X)  =  p-^ p-r    /  .  X'^ 

x^'+i  log.  X  x^'+i 

~     »i+  1  (m+l)2 


<§>  49.]  LOGARITHMIC    FUNCTIONS.  103 

Integration  of  logarithmic  functions. 


SO  that  the  required  integral  is 
m 


771+1       p  2  2         ~| 

-r-r    I   ('og.  x)2 ; — -  lo2.  x-\-- I. 


Secondly.  When  ?w  =  —  1 

(417)  gives  for  the  required  integral 

^(log,  x)3. 

2.  Integrate  x'"  (log.  x)^. 

Ans.  When  m  differs  from  —  1,  it  is 

a;m+l 


Lrno.     ..        3(log.x)^       3.  2.  log.  X         3.2.1-1 

1    L^      ^'       ^  ^«+3       ^(^/i+l)2  (m+l)3J 


W  -f- 

and  when  m  z=.  —  1 

it  is  I  (log.  x)*. 

3.  Integrate  f.  x.  log.  x. 

-471S.  When  jT.  x  differs  from  x~i,  it  is 

and  when  f.x^n- 

^  X 

it  is  J  (log.  x)2. 

4.  Integrate  -^— s — '-  when  w  differs  frOm  — 

An..  (}^^^. 

w  -|-  1 


(418) 


104  INTEGRAL    CALCULUS.  [b.  V.  CH.  TV, 

Logarithmic  definite  integrals. 

5.  Integrate  — — - — — -. 

"         (1— 2;)2 

X  lOff.  z 

6.  Integrate  — ^ .  Ans.  \os.^  x. 

^  X  log.  X  ^ 

50.  Problem.  To  find  the  value  of  the  definite  in- 
tegral 

n.(-\og.x)i  (419) 

in  which  n  is  an  integer  greater  than  — 2. 

Solution.  First.  In  this  case,  (408-410)  give 

/.  x=I,     F.x^^x,    f.f.xz=x',  (420) 

/.  (-  log.  x.Y  =  X  (-  log.  xf  +  ^f.  (-  log.  x)^-'  (421) 

in  which  when         x  =1  0,     or  =  ] ,  (422) 

the  first  term  of  the  second  member  vanishes  as  in  example  2 
of  B.  II.  §  109,  so  that  the  required  integral  becomes 

In-i-^og-^-'-  (423) 

By  this  process,  then,  the  exponent  of  ( —  log.  x)  is  dimin- 
ished by  unity ;  and  a  continued  repetition  of  it  gives  for  the 
value  of  (419) 

i(i-')(i-"") (l-''+^)f:-i-^o,..Mm 

n 
in  which  h  is  an  integer  not  greater  than  -  -f-  1. 


«5>  50.]  LOGARITHMIC    FUNCTIONS.  105 


Logarithmic  definite  integrals. 


Secondly.     When  n  is  even,  let 

h  —  ^n  (425) 

and  (424)  gives 

/i.(-Iog.2;)'^=1.2.3 h.  (426) 

Thirdly.    When  n  is  odd,  and  positive,  let 

h  =  ln  +  i,  (427) 

and  (424)  gives 

/■.(-log.^)'-*=  (A_J)(A_|)...|.J/..(_Iog..)-i.  (428) 
Fourthly.    When  n  =  —  1 

let  -^  =  /^(-log.2:)-i  (429) 

The  substitution  of 

xz=zay\  (430) 

in  which  a  is  supposed  less  than  unity,  so  that  (-—  log.  a)  is 
positive,  gives  ^ 

—  log.a:=:  —  y^log.a^  d^yXz^^ya^^.hg.  a;       (431) 

and  when  x  =  0,  y  ziz  cc  , 

xzzzl,    7/  =  0;  (432) 

K  —  —  2 /'^  a^\—  log.a)*,  (433) 

and  K{—\og.a)~iz=i—2f\a^\         (434) 

But,  by  taking  the  integrals  relatively  to  a,  we  have 

r\{-]og.a)-i=K,  (435) 

•/    0 


106  INTEGRAL    CALCULUS.  [b.  V.  CH.  IV. 

Logarithmic  definite  integrals. 

r\a^^  =  -^-a^^-^'=-JL_,  (436) 

Jo  J/^+1  3/^  +  1 

and,  therefore,  the  integral  of  (434)  with  reference  to  a,  is 
by  (304) 

X2__2  Z**'.— 1— =  — 2(tan.[-i]0— tan.[-i]oo) 

=  —2{0  —  irr)  =  ^.  (437) 

or         K=n.(^\og.x)-i=f\-^-  =  ^/rr.    (438) 

51.  Corollary,    The  substitution  of  (438)  in  (428)  gives 

/M>og.ir*=i^2:^'±Dv..         (439) 

52.  Corollary.     The  substitution  of 

X  =  3/"*+^     or     log.  X  z=z  (w  -f-  1)  log.  y,         (440) 
whence  d,y.  2;  r=  (m  +  1)  3/"*  (441) 

in  (426  and  439)  gives ;  by  dividing,  in  one  case,  by  (w+1  )^+^ ; 
and,  in  the  other,  by  {m  -\-  1)^2^ 

sir  (log.  ^Y  =  -^'I'iyXr  ■■  (442) 

/.,.(,og..)-J  =  Li:^(^V^.      (443) 

2^(m  +  l)*+^ 

53.  Problem.     To  integrate 

F.{ef'-).{f.xYd,.f.x,  (444) 


<§»  55.]  LOGARITHMIC    FUNCTIONS.  107 

Exponential  integrals. 

Solution.     Let  1/  z=.  e/-%  (445) 

whence  d^y.  x  m  (e-^-^  d^.f.  x)-^  =  (y  d^.f.  x)-^       (446) 

Jog.  y  —f-'^^  (447) 

and  the  integral  of  (444)  is,  by  §  19, 

f-i'^oS-vY-^,  (448) 

which  may  be  found  by  §  48. 

54.   Corollary,     When 

w  =  0  (449) 

(448)  gives 

f.F.(ef-).d,.f.z^f.^.  (450) 

55.     Examples. 

1.  Integrate     e'^^\/(l  —  e^'^''). 
In  this  case  if 

f.x~  ax,     dj.  xz=La 
F.y=:  ^y^[\^y2) 
(450)  gives 

/.  e«W(l  — «^"")  =  i/V(l  — ^2) 

=  i^e«*\/(l— e'''^)  +  Jsin.[-i]e''^ 

2.  Integrate   e«^.  Ans.     ie''\  (451) 

3.  Integrate   xe'^^  Ans.     ( ^  j  e  *»  *. 


108  INTEGRAL    CALCULUS.  [b.  V.  CH.  IV. 


Potential  integrals. 


4.  Integrate   a*.  Ans. 


log.  a 
6.  Integrate    Sin.  {kx-\-a).       Ans.  ^  Cos.  {kz-\-a).  (452) 

6.  Integrate  Cos.  (kx-\- a).       Ans.  lSin.{kx-\-a).  (453) 

56.  Problem.     To  integrate 

f.  (Sin.  k  X,     Cos.  k  x).  (454) 

Solution.     Let 

y  :=  Sin.  A;  a;,     or     kx  =z  Sin.[-i]y  ;  (455) 

and  by  (127  and  143) 

Co3.kx=V(l+y2),    kd,,,.x  =  ^-^^^^',     (456) 

whence  the  integral  of  (454)  is,  by  §  19, 

/.J/.[y,  V(l  +  y=)].(l  +  y^ri  (457) 

which  can  be  found  by  §  31. 

57.     Examples. 

1.  Integrate   Sin.'"  kx  .  Cos.  kx. 
Solution.     In  this  case,  (457)  becomes 

r  y^  —  ^ —  =z ■ — - .  (458) 

^    T  ^7^-7  >.  Cos.'^+iZrz 

2.  Integrate  Cos.'^Arx.  Sin.  A;  a;.         ^ns.     - — ,    ,,  ,  . 

^  (7^4-  1)  A: 

3.  Integrate  Tang,  k  x.  Ans.    ^  log.  Cos.  k  x.    (459) 


«5>  57.]  LOGARITHMIC    FUNCTIONS.  109 


Potential  integrals. 


4.  Integrate  Cotan.kx,  Ans.     -\  log.  Sin.  yt  z.    (4G0) 

5.  Integrate  Sec.kx.  Ans.   ^  tan. M]  Sin.  A:  a;.  (461) 

6.  Integrate  Cosec  k  x.         Ans.     .^  lo2.  i    ~ ^ZT    I 

^  '*     ^    \Cos.kx+l)' 

or,  by  (140),  ^  log.  Tan.  ^kx.     (462) 

7.  Integrate  e'^^Sin.  ^z.  (463) 
Solution.     Since  by  (121  and  122) 

Sin.  ^•2;  =  J  (e^*  —  e-^'^), 
Cos.  ^z  :=  J  (e^x-j-e— Ax)  . 

we  have        e  '^  ^  Sin.  ^  a:  =  J  (e('^+^)^  —  e^'^-^')^), 

f.e'^^Sin,kxz=^ — 

^  2{a+k)       2{a—k) 

\  ^{a2—k2)  } 

^^  /a  Sin.  kx  —  k  Cos.  k  x\ 
=  '     (  ,i^^p )•    (464) 

8.  Integrate  e"*  Cos.  Arz. 

An^      .aJa^os.kx—kSm.kx\ 

Ans.     e    y  a2-k2       ^   )•  (46o) 

9.  Integrate  c"*  Sin.  a  a;.  Ans,     i^e'^"^  —  J  z.     (466) 

10.  Integrate  e '' "^  Cos.  a z.  Ans.     ^e'^'"' -\-^x.     (467) 

11.  Integrate  e^-^'^^-^'^Cos.kx.         Ans.    ^€«sin. Az^  (468) 

12.  Integrate  c«^o^-*^  Sin.  ^  x.         ^ns.  ^.  e "^  Cos.  A: a;^   (469) 

10 


110  INTEGRAL    CALCULUS.  [b.  V.  CH.  IV. 

Potential  integrals. 

68.  Corollary.    The  differential  coefficients  of  (451,  452, 
453,  458,  4C4,  465)  with  respect  to  a,  A-,  or  m,  give 

/x^e'^'  =rf^.,.ie'^*;  (472) 

/  xCos.  {kx  +  a)  z=z  d,,,.  i  Cos.  (kx  +  a)  (473) 

X  1 

z:^jS'm.(kx-\-a)  —  —  Cos.  (A;  x  +  «) 

f.x^Sin.(kx  +  a)=idl,.lCos.(kx-{-a)  (474) 

=  ('^  +  i)^os.(kx+a)^^^Sin.{kx  +  a), 

/  x2«+iCos.(A;x  +  a)  =  dl",+\  i  Cos.  (k  x  +  a),  (475) 

/.  x^"  Sin.  (A:x  +  a)  z=  d^J.  i  Cos.  (A:x  +  a),  (476) 

/  x2«+i  Sin.(Z:  X  +  a)  =  4.-  ^L"+^  i  Cos.  (A;x  +  a) 

z=c/?;,+i.i  Sin.(A;x  +  «),  (477) 

/.x^^Cos.  (A;x+a)  =  c?.M  Sin.  (kx  +  a),  (478) 

Sin.™+1  A:x 
f.  Sin.*"  A;  x .  Cos.  k  x .  log.  Sin.  kx=:  c?,.^.    ,  (479) 

(1     \  Sin.^'+i  k  X 
log.  Sin.  kx j— -  I  7—{VT  y 


§  59.]  LOGARITHMIC    FUNCTIONS.  HI 


Potential  integrals. 


f.Cos.-H.Sin.kx.log.Cos.kx  =  d„„.^''^     (480) 


=  I  loff.  Cos.  Jcx  — I 


Cos."'+^  k  X 


—  Ix—  ^"^    \.-  aSm.kx-kCos.kx   .   c-Sin.Arz  ,,^^^ 

/x;e-Sin.^.==^.,.(e^^^^^i!^=|^^  (483) 

59.   Corollary.    Equations  (121  and  123)  give 

x"  e-^*  Sin.  «  a;  =  J  z"  e^ax^.  j  ^n^  ^^g^^ 

2"  c'^^Cos.  a  a;  =z  J  a;"  c'-^^  +  ^z'' ;  (435) 

so  that  by  (472) 


(486) 


/  x"  c- Sin.  a  a:  r=  ^  c/^.^.  i- c^- -  _L^  a;«+i 

2a  2(?i-f-l) 

2n+i     —2^^  2(w+l)^      ' 

/x"e-Cos.ax=ic/,.2..l62-  +  ^^-l_  (487) 

2n+i«c.«.2^«      +2(n+l)'^      • 


112  INTEGRAL    CALCULUS.  [b.  V.   CH.  IV. 


Exponential  definite  integrals. 


60.  Corollary.    When 

n—  1 


(486  and  4S7)  become 

/  2  c«^  Sin.  a  z  =  J  J,., .  ji  e2 '^^  —  ^  x2  (488) 

/  X  e«»  Cos.  a  I  =  ^  rf,.  .  i;  c2"  +  J 12  (489) 


—      A       »> 


1    /.'-ia* 
4 


61.  Problem.  To  find  the  value  of  the  definite  inte- 
gral 

J».3.ng_ax2^  (490) 

in  which  a  is  positive  and  n  is  zero  or  a  positive  in- 
teger. 

Solution.     Let 

6-^^  =  2/,     c"'=:^,  (491) 

x2  —  _  log.  3/  =  log.  i ,  (492) 

2xd,,,.x  =  -i,     d^^^.x  =  -^;  (493) 

so  that  when                 x  r=  0,     ?/  =  1,  (494) 

X  =  OD,    y  =  0.  (495) 
The  value  of  (490)  is,  by  §  19, 

4/^0og.^)^r-'.  (496) 


<§,  64.]  LOGARITHMIC    FUNCTIONS.  113 

Exponential  definite  integrals. 
Hence,  by  (442  and  443),  when  n  is  odd, 

.  ^  -a^^  1.2.3  (^n-i)_ 

and  when  n  is  even 


TC 


/.^      „  —az^       1.3.5 in-X-V)      ...  I  Ac\Q\ 

/; .  x" e  = j^^ — ^ — '-  V— .  (498) 

2  (2  a)*"  "^ 

62.  Corollary,     When 

w  =:  0,       a  :=  1, 
(498)  gives 

/«.e-"'=iV-.  (499) 

63.  Corollary.  By  reversing  the  sign  of  x  (497  -  499)  give, 
when  w  is  odd, 


ax2_       1.2.^...(^w  — ^) 

2a^ 
when  »  is  even 


y_o..,.e  = —^^ ^,  (500) 


2(2a)^"  " 


/_o„.e-^  =  ^V-.  (502) 

64.  Corollary.    The  sums  of  (497  and  500),  of  (498  and 
501),  of  (499  and  502),  give,  when  n  is  odd, 

/_"«.x"e-^^'=0,  (503) 

10* 


114 


INTEGRAL  CALCULUS.      [b.  V.  CH.  IV. 


Exponential  definite  integrals. 


when  n  is  even, 


J-co-^  e  =  — V— ,  (504) 

(2«)^"  " 


/-"oo.c-^  =-s/n,  (505) 


G5.  Corollary.     The  substitution  of 


.  +  ^„  (506) 

for  X  in  (504)  gives  by  §  18,  when  n  :rz  0, 

/f„.  e-("'+*^+ 4-a)^  ^iL  ;  (507) 

which,  multiplied  by  e        4a ^  gives 

/_"a,.e-(^^'  +  *^  +  ^)=e47     '  ^IL,  (508) 

66.  Corollary.    The   differential  coefficients  of  (50S),  with 
reference  to  a  and  6,  are 


67.    Examples. 
1.  Find  the  value  of  the  definite  integral 
/_"„ .  a:  e-  Ca  ^2  +  c)  gin.  A;  x. 

MS.   ^^e 


<5>  67.]  LOGARITHMIC    FUNCTIONS,  115 


Potential  definite  integrals. 


2.  Find  the  value  of  the  definite  integral 

/_"oo .  X  c-  («^^  +  '=)  Cos.  k  X.  Ans.     0. 

3.  Find  the  value  of  the  definite  integral 

/-"oo  •  {m  »2  +  n)  e-  («  ^'^  +  ^)  Sin.  k  x,         Ans.     0. 

4.  Find  the  value  of  the  definite  integral 

/_*«, .  (m  x2  4-  n)  e-  ^^  *^  +  <=)  Cos.  k  x. 

Ans.     (^  +  -^+Ae^«~Vf 
V  4a2~2a  '      /  « 

5.  Find  the  value  of  the  definite  integral 

r  a>    p —  a  X 
J    0**^  i 

in  which  a  is  positive.  Ans.     ^.  (511) 

6.  Find  the  value  of  the  definite  integral 

in  which  a  is  positivet 

-  1 . 2  . . . .  7t 

^''''  an  +  i    -'  (512) 


116  INTEGRAL    CALCULUS.  [b.  V.  CH.  V. 


Trigonometric  integrals. 


CHAPTER   V. 

INTEGRATION    OF    CIRCULAR    FUNCTIONS. 

68.  Problem.     To  integrate 

f.  (sin.  k  X,  COS.  k  x).  (513) 

Solution,     The  substitution  of 

xzzryV— ^     <^c.y.a:  =  \/— 1,  (514) 

gives  by  (121  and  122) 

^v[i.kxz=z\/ — l.Sin.  A;y,  (515) 

COS.  kx  z=.  Cos.  k  y  ;  (516) 

and  the  integral  of  (513)  is 

y:^_-l./.(v'— 1.  Sin.Ajy,  Qos.ky),        (517) 
which  may  be  found  by  §  ^Q. 

69.     Examples. 

1.  Integrate  sin."*  kx.  cos.  kx. 
Solution.    In  this  case  (517)  becomes 

(_  1 )? ('"+^^  ^'m.^ky.  Cos.  k y  ; 
whence  by  (458,  121,  122,  and  514) 


^  69.]  CIRCULAR    FUNCTIONS.  117 


Trigonometric  integrals. 


^  ^        ^  {m+l)k 

sin.'"  +  i  kx 


(m+  l)k' 
2.  Integrate  cos.'"  k  x  .  sin.  k  x. 


(518) 


COS."*  + 1  A:  2: 

^"^- 7 r-T-T--  (519) 

(wi  -f  1)  A;  ^        ' 

3.  Integrate  sin.  (kx  -\-  a). 

Ans.     —  i  COS.  (A;  I  +  a).     (520) 

4.  Integrate  cos.  (k  x  -\-  a). 

Ans.     ^  sin.  (kx  -{-  a).     (521) 

5.  Integrate  tang.  A:  2.  Ans.     — -^  log.  cos.  A:  a;.     (522) 

6.  Integrate  cot.  A- z.  Ans.     -^  log.  sin.  A:  2 .     (523) 

7.  Integrate  sec.  k  x.  Ans,  ^  log.  .  '    -.     (524) 

1.  — —  Sill*  /l  3/ 

8.  Integrate  cosec.  k  x.         Ans.    ^  log.  tang.  J  k  x.     (525) 

9.  Integrate  e°*sin.  kx.  ^526) 
Solution.  The  substitution  of  (514)  in  (526),  gives  by  (464), 

/  e'^^sin.  A:x=  — /6«J/^^-i  Sin.  ky 

as/ — 1  Sin.  k  y — k  Cos.  k  y 

—  _(a2_j_y^-2) 

_  ^^  ,  flsin.  A:x~A:cos.A:x 

^2   _|_   ^2 


118  INTEGRAL    CALCULUS.  [b.  V.  CH.  V. 


Trigonometric  definite  integrals 


10.  Integrate  e***  cos.  kx. 

a  COS.  k  X -\- k  ^'\n,  h  X  ,rc\a\ 

Ans.     e- ^^ .        (528) 

11.  Integrate  c«s'"-'^^  COS. /ex.  ^?is.  J*  c**"'"-*^.     (529) 

12.  Integrate  e<'^^^-^^  sm.kx.         Ans.  —^,e<'^^^-^\  (530) 

70.  Corollary.    The   differential  coefficients  of  (518-521, 
627,  528),  with  reference  to  m,  k,  and  a,  give 

f.  sin.'"  k  X .  cos.  k  x .  log.  sin.  k  x 

/i         •      7                1      \sin.»»+iA;x 
=  i  log.  sin.  A;  X ttIt TTTZ'     (^'^^/ 

f,  COS.'"  kx,  sin.  k  x.  log.  cos.  k  x 

=    I  r-T  —  log-  COS.  kx\ r-zrr-r       (532) 

\7?i  +  l         ^  /(m  +  l)A:'    ^       ' 

f.xcos.  {kx-]-a)z=i— dc.k'^ COS.  {kx  -\- a)  (533) 

1  a^ 

=  —  COS.  (^  2^  +  «)  +  T  sin.  (A;  X  +  a), 

(__l)»y:x2'»sin.  (ytx  +  a)=-~  </,'.?.  i  cos.  (A;x  +  «),  (534) 

(__!)"/ x2«+icos.(A:3;  +  a)-—<^.^+i.i  COS.  (A;x  +  o),  (535) 

(— l)"/z2"  COS.  {kx  +  a)  =       dll.  i  sin.  (kx  +  a),  (536) 

(__l)"y:x2«+isin.(/i;x+«)  =  —  dl",+Klsm.{kx-^a);  (537) 

a  sin.  A;  X — A;  cos.  A:  x 
f,xe'sm.kx=:  d,,,.  c''^ ^Fipp 

(2a     \asin.A:x-A;cos.A:x   ,    e'*'sin.  A:x      .^no\ 
^-^qrp)     «2  +  fc2 — +^?+i^'  (538) 


^  72.]  CIRCULAR    FUNCTIONS.  119 

Trigonometric  definite  integrals. 

.     ,           7            a  sin.  A;  X — k  cos.  lex         ,^^r^^ 
/2;"e°^sin.  kx=i  dl,.  e^ -^    ,    ^^ ,        (539) 

a  COS.  ^  z  +  A:  sin.  ^  X         ,^.r^^ 
/  x" e-^ COS. k%—  c?«, .  e ^ ^F4rp •       (^^^) 

71.  Problem.  Find  the  value  of  the  definite  integral 

/f„.  (mx2  +  w)e-'^^^  +  '' COS.  A:x.  (541) 

Solution.    The  substitution  of  k\^ —  1  for  k  in  example  4 
of  §  67}  gives  for  the  value  of  (541) 

(mk^    ,    m    .      \ 
■""  1 T^  +wl  e 
4a     '  2a         / 


47-'.  ^iL.       (542) 

a 


72.     Examples. 

1.  Find  the  value  of  the  definite  integral 

/.*,  xe-Ca^'^  +  c)  sin.  k z. 

_  k2 

Ans,   ~le     ^      V-^.    (543) 
2a  ^ 

2.  Find  the  values  of  the  definite  integrals 

/.=°„.  xe-(«^'^+'')  COS.  kx, 
and  /."„.  (m  x2  -|-w)  e-(ax2+c)  gju^  ;^2.^ 

Ans.     0.       (544) 

3.  Find  the  value  of  the  definite  integral 

/S".  e-«^sin.  A;x,  (545) 

in  which  a  is  positive. 


120  INTEGRAL    CALCULUS.  [b.  V.  CH.  V. 

Trigonometric  definite  integrals. 
Solution.  Equation  (527)  gives,  for  the  value  of  this  integral, 

4.  Find  the  value  of  the  definite  integral 

/^.  c-°^  COS.  A:z.  Ans.  ■  .    (547) 

73.  Prohlejn.  Find  the  value  of  the  definite  integral 

in  which  w,  w,  and  ^  are  positive  integers. 
Solution.     The  substitution  in  (548)  of 

sin.  x=z  y,     z  =  sin.[-i]  3/ ;  (549) 

whence         cos.  2;=:  (I — 1/^)^ ,  d^,..  x  =z  {I — y^)  ^  ;     (550) 

and  when           x  =z  0,      y  =  0,  (^^1) 

x  =  {2n  +  i)rr,     y=l;  (552) 
gives  for  the  value  of  (548) 

fi.y(l-y^)i(j'-l),  (553) 

which  may  be  found  by  §  44. 

74.     Examples. 
1.  Find  the  value  of  the  definite  integrals 

'(2  71  +  ^)  TT 


.  sin.""  X 
0 


f 

/•(2n  +  i)7r 

and  /  .  cos.'"2 


when  m  is  an  integer. 


§  74.]  CIRCULAR    FUNCTIONS.  121 

Trigonometric  definite  integrals. 

.         „^,  .      ,,   .    .    2.4 (m  —  1)     ,^^^, 

Alls.    When  m  is  odd,  it  is  — ^ (554) 

3.5 m  ^       ' 

1  1.  o (wi— 'ij   ._       ,-v  <>.».,v 

when  m  is  even,  it  is (2  w  +  J)  n,        (555) 

2.  Find  the  value  of  the  definite  integral 

/(2n  +  i|)7r     .  ,.^-. 

.  sm.  % .  cos.  X.  (556) 

Ans.     1. 

3.  Find  the  value  of  (548),  when  m  and  p  are  both  even. 

1.3.5 (m-l)X1.3.5...(p-l) 

'•         2.4.6 (ro  +  i)) (2«+J)=^-(557) 

4.  Find  the  value  of  (548),  when  m  is  even  and  p  odd. 

1.3.5...(«-l)X2.4.6....(j>-l) 
'•  17375 ('«+^  ' 

2.4.6.  ..(j)  — 1) 
°'       (»»+l)(»,  +  3)....(»t+^)-  (^^^) 

5.  Find  the  value  of  (548)  when  m  is  odd,  and  p  even. 

2.4  6....(m-l)Xl.3.5....(p-l) 
1.3.5 (»»+?)         ' 

2.4.6.....(m  — 1) 

"■^      (P  +  l)(i'  +  3)..('«+i'r  ^       ' 

6,  Find  the  value  of  (548),  when  m  andp  are  both  odd. 

An.  i^-4-6^-(»-l)X2.4.6.     (y     1) 
2.4.6 (m  +P) 

or  the  same  with  the  second  answers  in  (558  and  559), 
11 


122  INTEGRAL    CALCULUS.  [b.   V.    CH.  V. 

Trigonometric  definite  integrals. 

75.  Problefu.  To  find  the  value  of  the  definite  inte- 
gral 

.  sin.'"  a:  cos/  x, 
in  which  m,  n,  and  p  are  positive  integers. 

Solution.   The  reduction  may  be  made  in  this  case,  precise- 
ly as  in  ^  73,  it  being  observed  that  when 

x=zO,     or  =2  Tin,     y  =  0.  (561) 

By  this   means,  the   integral,  when   either  m  or  p  is  odd,  is 
zero  i  but,  when  m  and  p  are  both  even,  it  is 

135      (,»-!)  X1.35...^,,l)^^^ 
2.4.0 {"^-j-p) 


76.     Examples. 
1.  Find  the  values  of  the  definite  integrals 

«/    0 

2  71  TV 


,  sm.'"  X 

0 


/.  cos/"  X 
0 

when  m  is  even. 


1.3.5....(m— 1)  ^  ,.^„, 

^^^S'         ^    ,   ^  — 2nTv.  (563) 

2 . 4 . 6 . . .  .    wi  ' 


4.  Find  the  value  of  the  definite  integral 

.9 


.  sm.  n  X .  cos.  k  a:, 
when  h  and  Jc  are  integers.  A7is.     0.  (564) 


^   76.]  CIRCULAR    FUNCTIONS.  123 

Trigonometric  definite  integrals. 

3.  Find  the  values  of  the  definite  integrals 


/  .  COS.  h  X  .  COS.  k  Xy 

/2w7r      .       ,  .       , 

.  Sin.  h  X,  sm.  A;x, 


v.'hen  h  and  k  are  integers. 

Ans.     It  is  zero,  unless         h  3=  k, 

in  which  case,  it  is  n  ^r,  (565) 


124  INTEGRAL    CALCULUS.  [b.  V.   CH.  VI. 

Length  of  the  arc  of  a  curve. 


CHAPTER   VI. 


RECTIFICATION    OF    CURVES. 


77.  Problem.  To  find  the  length  of  an  arc  of  a 
given  curve. 

Solution.  If  s  denotes  the  required  arc,  its  length  is  readily 
found  by  ascertaining  the  value  of  its  differential  coefficient, 
and  integrating  it. 

Thus  if  we  adopt  the  notation  recently  introduced  by  some 
of  the  most  eminent  mathematicians,  and  denote  the  differen- 
tial coefficient  by  the  capital  letter  D,  and  denote  by  a  small 
letter  annexed  to  D  or  /,  the  corresponding  independent  vari- 
able, we  have  by  (570-582  of  vol.  1), 

s  =fDs  =/.  V  [I  +  (^.3/^)  =f:c '  sec.  T  =  /,  .  cosec.  r 

=frV  [r'  +  [D,,  rf]   =fW[^  +  r'  (A  ^ 

=zf^ .  sec.  £  =zf'(p.i-  cosec.  s.  (566) 

76.  Corollary.  The  arbitrary  constant,  which  is  to  be 
added  to  complete  each  of  these  integrals,  corresponds  to  the 
indeterminateness  of  the  point  at  which  the  measured  arc 
may  commence.  The  condition,  by  which  this  point  may  be 
determined,  will  be  sufficient  to  determine  the  value  of  the 
arbitrary  constant;  or  to  eliminate  it  and  reduce  the  result  to 
the  form  of  a  definite  integral.     Thus,  if  the  length  of  the 


§  78.]  RECTIFICATION   OF    CURVES.  125 

Arc  of  hyperbola  and  cycloid. 

arc  is  required,  which  extends  from  the  value  of  Xq  to  that  of 
X  J,  it  is  evidently  represented  by  the  definite  integral 

5  =/^^  D  s.  (567) 

78.    Examples. 

1.  Find  the  length  of  the  arc  of  the  curve  of  which  the 
equation  is 

Solution.     In  this  case,  we  have 

D  s    =  J  (g^  +  e-^) 
s   =  J  (e^  —  e-^), 
in  which  the  length  of  the  curve  vanishes  with  x  =  0. 

2.  Find  the  length  of  the  arc  of  the  parabola  whose  equa- 
tion is  1/^  z=2  p  Xi 

counted  from  the  vertex.  (568) 

3.  Find  the  length  of  the  cycloid  from  equations  (130,  13J, 
of  vol.  1),  the  arc  being  supposed  to  commence  with  x. 

Ans.     4  i?  (1  —  cos.  i&)=S  R  sin.^  J  &,         (569) 

and  the  whole  length  of  a  branch   is  8  R,  corresponding  to 

^  =  2  ^.  (570) 

11* 


126  INTEGRAL    CALCULUS.  [b.   V.    CH.   VL 


Arc  of  hyperbolic  and  logarithmic  spiral. 


Ans.     r  sec.  « = 


4.  Find  the   length   of  the   liyperbolic  spiral,  the  arc  being 
supposed  to  commence  with  (p  r=  (Pq.  (5^1) 

.„.«[v(..L)-v(.+,y+%(rf;^3)]- 

5.  Find  the  length  of  the  logarithmic  spiral,  the  arc  being 
supposed  to  commence  with  r. 

The  elliptic  and  hyperbolic  arcs  possess  some  peculiar  prop- 
erties, which  deserve  particular  investigation. 

79.    Theorem.     The  two  tangents^  which  are  drawn 
from  a  given  point  to  a  given  ellipse  or  hyperbola,  make 
equal  angles  with  the  two  lines  which  are  drawn  from 
the  same  point  to  the  tivo  foci. 

Thus  the  two  tangenti>  P  T  and  P  T'  (figs.  1,2,3),  make 
equal  angles  with  the  lines  Pi^  and  P  F'  drawn  to  the  foci; 
that  is,  the  angles  F P  T  and  F'  P  T'  are  equal. 

Proof.  Each  of  the  two  tangents  P  T  and  P  T'  is,  by  ex- 
amples 2  and  3  of  §  131  of  vol.  1,  equally  inclined  to  the 
lines  drawn  from  the  foci  F  T  and  F'T.oxF'T  and  F  T', 
so  that  the  angles 

F  Tt  =  F'  TP ,  and  F  T'  P  —  F'  T'  t'. 

If  then  the  triangles  FTP  and  F'  T '  P  ^xe  turned  over, 
around  the  sides  T P  and  T'  P,  which  remain  stationary,  so 
as  to  fall  into  the  positions  TP S  and  T'  P  S\  the  points 
S  and  >S''  will  be  in  the  lines  T F'  and  T'  F  produced  if  neces- 
sary. The  triangles  P  SF'  and  PS'F  are,  then,  equal; 
for  the  sides  P  S  —  P  F, 

P  F'  —  PS'  ; 


<§>  81.]  RECTIFICATION    OF    CURVES.  127 

Elliptic  and  hyperbolic  arcs. 

and  the  side  i^' >S^  ^  i^>S^',  because  each  of  these  two  lines 
is  equal  to  the  transverse  axis,  since  in  (fig.  1)  each  is  the  sum 
of  the  two  lines  F  T  and  F'  T,  or  of  the  two  F  T'  and  F'T'; 
while  in  (figs.  2  and  3)  each  is  the  difference  of  the  same  two 
lines.  The  angles  S P  F'  and  F P  S'  are  consequently  equal. 
If  the  angle  FP  F'  is  subtracted  from  each  of  these  angles 
(fig.  1),  or  added  to  each  of  them  (fig.  2),  or  diminished  by 
each  of  them  (fig.  3)  ;  the  resulting  angles  SP  F  and  S'  P  F' 
(figs.  1  and  3),  or  the  excess  of  360°  over  ,SrPF  and  S'  P  F' 
(fig.  2)  are  equal.  Hence  F  P  T  and  F'  P  T',  which  ar^ 
the  halves  of  S  P  F  and  >S"  P  F',  are  equal. 

80,  Corollary.  If  an  ellipse  (fig.  1)  or  an  hyperbola  (figs. 
2,  and  3)  be  drawn  with  the  points  F  and  F'  for  foci,  and 
passing  through  the  point  P,  the  tangent  to  this  new  curve  at 
the  point  P  will  be  equally  inclined  to  the  two  lines  P  F  and 
PF';  and,  therefore,  it  will  also  be  equally  inclined  to  the 
two  tangents  TP  and  T  P. 

81.  Theorem.  If  froin  any  'point  of  the  ellipse  PP' 
(fig.  1),  or  of  the  hyperbola  P  P'  (fig.  2),  ivhich  has 
the  points  F  P'  for  its  foci^  tangents  are  drawn  to  the 
ellipse  T  T'  or  hyperbola  T  T'  lohich  has  the  same  foci, 
the  sum  of  the  tangents  P  T  and  P  T'  exceeds  the  in- 
cluded arc  T  T'  by  a  constant  quantity  ;  that  is,  by  a 
quantity  which  is  the  same,  from  whatever  point  of  the 
first  ellipse  or  hyperbola  the  tangents  be  drawn. 

Proof.     Let  tangents  p  t" ,  p  t'   be    drawn    from   a  second 

point  p  infinitely  near  P.     The  tangent  yt"  exceeds  P  T  hj 

the  projection  of  Pp  upon  P  T  diminished  by  the  arc  Tt", 

or 

pt"  =  P  T-\-  P  P'  cos.  TP  S—  Ti", 


128  INTEGRAL    CALCULUS.  [b.   V.   CH.  VI. 

Elliptic  and  hyperbolic  arcs. 

In  the  same  way, 

2)i'=:PT'  —  P  P'  COS.   T'PS'+T'  t', 

z=  P  T'  —  P  P' COS.   TPS-}-  T'  t'-, 

whence 

jpt"+2't'  =  PT-\-P  T'  +  (T'  t'  —  Tt"). 

But  t"t'  =  T  T'  -{-{T'f  —  Tt"); 

and,  therefore, 

(j)t"  +  p  t')  —  t"  t'  =  {P  T  +  P  T')  —  T  T'. 

The  excess  of  the  sum  of  the  tangents  over  the  included 
arc  does  not,  then,  increase  by  moving  the  point  P  a  small 
distance  upon  the  curve,  in  which  it  is  situated ;  and  conse- 
quently this  excess  must  be  a  constant  quantity. 

82.  Corollary.  Had  an  hyperbola  P  Q,  (fig.  1),  or  an 
ellipse  P  Q  (fig.  2),  been  drawn,  with  the  foci  jP  and  P',  it 
might  easily  have  been  shown  in  the  same  way,  that  the  excess 
of  the  difference  of  the  tangents  P  T  and  P  T'  over  the  dif- 
ference of  the  arcs  Q  T  and  Q  T'  was  constant.  But  as  the 
point  P,  in  moving  along  the  curve  P  Q,  approaches  Q,  the 
tangents  and  arcs  decrease,  and  finally  vanish  when  P  coin- 
cides with  Q.  At  the  point  Q,  therefore,  the  excess  of  the 
difference  of  the  tangents  over  the  difference  of  the  arcs  is 
nothing,  and  therefore  this  excess  is  nothing  for  every  point  of 
the  curve  P  Q. 

Hence,  if  from  any  'point  P  of  the  hyperbola  P  Q 
(fig.  l)j  or  of  the  ellipse  P  Q  (fig.  2),  which  has  the 
points  F  a7id  F'  for  its  foci ,  tangents  are  drawn  to 
the  ellipse  T  T'  (fig.  1),  or  to  the  hyperbola  T  T' 
(fig.  2)j  which  has  the  same  foci^  the  difference  of  the 
tangents  P  T  and  P  T'  is  equal  to  the  difference  of  the 
arcs  q  T  afid  q  T'. 


^84.] 


RECTIFICATION    OF    CURVES. 


129 


Elliptic  arc. 


83.  Corollary.  If  the  excess  of  the  sum  of  the  tangents 
P  r  and  P  y  over  the  arc  TT'  is  denoted  by  2  E,  the  two 
preceding  theorems  give 

PT+PT=QT-\-QT'  +  2E 
F  T'  —  P  T  =  Q  T—  Q  T; 


whence 


PT'  =  QT'  +  E 
PT=  QT-\-  E. 


(573) 


84.  Corollary.  Upon  the  transverse  axis  A  A'  (fig.  4)  of  the 
ellipse,  describe  the  semicircumference  ^li  Z.'^',  dravy^  the 
ordinates  L  T  31  and  L'  T'  M',  and  join  O  L,  O  L',  O  being 
the  common  centre  of  the  ellipse  and  circle.  Let,  if  O  5  is 
the  semiconjugate  axis, 

if  z:^  LOB,  tp'  z=  L'  OB, 

A=OA     ,  B=OB, 


x  =  OM  , 

!/  =  MT  , 

z  =  ML  , 
sz=:B  T 


%'  =  OM', 
y>  =  M'  T\ 
%'  —  M'  L', 
s'  =zB  T', 


>     (574) 


1       ^' 


we  have,  by  section  163  of  vol.  1,  and  by  the  triangles  LOB, 
LOB, 


x-=zA  sin.  <p. 


%'  =  A  sin.  y', 


(575) 


7?  7? 

y^n  —  .  z  =  -  ,  A  cos.  <pz=:  B  COS.  <p,    y'=B  cos.  (p' ;  (576) 

JL  -A. 


130  INTEGRAL    CALCULUS.  [b.  V.    CH.   VI. 

Elliptic  integral  of  the  second  order. 

and,  by  differentiation,  letting  ^  be  the  independent  variable, 
D  X  :=:  A  COS.  T,       D  1/  =  B  sin.  y.  (^^^) 

Ds^  =  A^  C0S.2  <p  -f-  J32  sin.2  cp 
=  A^ -\- (B^  —  A^)  sin.2y 
=  ^2  (1  __e2sin.2T),  (578) 

Ds=A^{i—e'  sin.2  g.) ;  (.579) 

s=fA^{l—e'^  sin.2  ^ ),  (530) 

This  integral  is  one  of  a  class  which  are  called  elliptic  inte- 
grals. There  are  three  integrals  in  this  class,  and  the  present 
one  is  said  to  be  of  the  second  order.  The  following  notation 
has  been  universally  adopted. 

J  [e^)z=z  a/(1  — c2sin.2  9',)  (581) 

or  the  ^  may  be  used  without  the  e  (p,  when  there  is  no  danger 
of  confusion, 

E  .  ^  ^fl  J,  (582) 

^^  E  .{e  cp)  —f'^  ^  {e  9).  (583) 

Hence  s  =  A  E  .  ^p.  (584) 

85.  Corollary.  Let  the  two  tangents  L  R  and  L'  R,  drawn 
to  the  circle  at  the  points  L  and  L',  meet  at  R.  By  reducing 
all  the  ordinates  of  the  circle  in  the  ratio  of  jB  to  A,  the  circle 
is  changed  into  the  ellipse.  By  the  same  system  of  reduction, 
the  lines  R  L  and  R  IJ  will  be  changed  into  other  straight 
lines  P  T  and  P  T',  and  these  will  be  tangent  to  the  ellipse, 
because  the  points  T  and  T'  are  the  only  ones  whose  ordinates 
vi^ill  be  as  small  as  the  corresponding  ordinates  of  the  ellipse. 
By  joining  O  R,  the  right  triangles  O  R  L,  O  R  L',  and 
ORN  give 


<§>  85.]  RECTIFICATION    OF    CURVES.  131 

Elliptic  integral  of  the  second  order. 

^^== — T^  =  — i^ — Tr>  (-585) 

COS.  LOR       COS.  h  {'P —  f/  ) 
ON=  ORcos.RON='^''"-}y  +#,        (586) 

COS.    ^    {'p  <f'') 

RN=OR  sin.  RON=  Aco^ii'r  +  'f') 

COS.  h  {V  —  y  ) 

PN=?.RN=^^^Ul±p.  (587) 

^  COS.  ^  {(p  (^') 

The  condition  that  the  point  P  is  upon  an  ellipse  or  an 
hyperbola,   as  in  fig.  1,  of  which  the  semiaxes  are  A'  and  B, 
gives,  by  putting 

where  the  upper  sign  corresponds  to  the  ellipse  and  the  lower 
to  the  hyperbola,  the  equation 

a  sin.^  i  (y  +  ./)        6  cos.^  H^/^  +  ^0^  ,  ,.oo. 

C0S.2  ^  ((f  _  9)     "f"     COS.^  i   (y  —  V')  '  ^       ^ 

or 

a  sin.2  i  (ff  +  cp')-|-6  cos.2  J  (^+9')=cos.2  j  (9)--^/),  (590 ) 

and 
a — fl  COS.  ((p-\-(p')-\-b-\-b  COS.  {(p-\-(p')=:l-\-cos.  {(p — <p')f  (591) 
a-\-b  —  l  =  {a  —  h)  cos.  {'p  -\-  q')  -\-  cos.  (t  —  (p') 

=:(a-b-{-l)  COS.  (p  COS.  f/''-|-(5-a-[-l)  sin.  9  sin.  9^^'.  (592) 

If  the  point  P  were  taken  so  that  T"  coincided  with  B, 
(f'  would  be  zero,  and  if  the  corresponding  value  of  y  is  de- 
noted by  (pQ ,  (59"2)  gives 

a+b  —  \  =  {a  —  b  +  \)  COS.  cp^  ;  (593) 


132  INTEGRAL    CALCULUS.  [b.  V.   CH.  VI. 

Elliptic  integral  of  the  second  order. 

which,  substituted  in  (592),  divided  by  a  —  h  -\-\,  gives 

,   ,   h  —  a  -\-  \     . 

COS.  </'o  =  COS.  y  COS.  <f'  -J sin.  v. sin.  </>'.  (594) 

a,  —  0  -j-  1 

But  the  condition  that  the  given  ellipse  has  the  same  foci 
with  the  curve  in  which  P  is  situated,  gives 

A^—B^  =zA'^^B'^=z——^,  (595) 

a  0 


Hl-'hHl-')' 


(596) 


,.^1       ^^-,     Ml-«)  _      a-h 

A^~       a  (1—6)  "~«(1— 6) 
But,  by  (593), 

/«+6— 1\2      4  a  (1—6)  ,^^^, 


sin. 


2 


4  (a  —  J) 


,     _  {b-a+lY  ^      _  .   b-a+l  .         ,503. 

^^0  -  (a_6+l)2    '        ^^0  -±  ^ZZj+r-        (^^^) 

which,  substituted  in  (594),  gives 

COS.  ({q  =  COS.  q;  COS.  (p'  i  sin.  y)  sin.  ^'  ^  .  g)^.     (599) 

Arid  this  equation  of  condition  is  the  same^  with  the  con- 
dition that  the  poiiit  P  of  fig.  1  is  upon  an  ellipse  or 
hyperbola  having  the  same  foci  with  the  given  ellipse^ 
the  upper  sign  of  (599)  corresponding  to  the  ellipse^ 
and  the  lower  to  the  hyperbola. 

86.  Corollary,  If  a  spherical  triangle  (fig.  5)  be  drawn,  of 
which  the  sides  are  9,  H>'  and  </>q  ,  and  the  opposite  angles 
^»  ^'i  ^0  »  ^^®  ^^^^  ^y  (3^1)  °^  Spherical  Trigonometry, 


§  87.]  RECTIFICATION    OF    CURVES.  133 

Elliptic  integral  of  the  second  order. 


COS.   (fin  COS.  (p  COS.   m' 

COS.  flo  = '-^—. .-^-- '—  ; 

sin.  (p  sin.  (p' 

(600) 

whence  by  (599), 

COS.  5o  =  ±:  ^  To  . 

(601) 

Hence 

C0S.2  ^Q  rz:  1  —  e^  sin.2  (p^  , 

(602) 

esin.  (Pq  =z  \/  (1 — cos.^  &^)  =z  sin.  dy  ; 

(603) 

and  therefore 

sin.  &Q         sin.  6       sin,  6' 

r604^ 

sin.  (pQ         sin.  (p       sin.  (p' 

This  equation  gives 
Jif=zj^{\—e^  sin.2  9')=\/(l— sin.2  ^)==Fcos.  5,   (605) 
^^'^^(l— e2  sin.2  ip')—  ^(l— sin.2  &')z=z      cos.  ^'.    (606) 

The  upper  signs  in  (601)  and  (605)  correspond  to  f 
the  case  in  which  the  curve  is  an  ellipse,  and  the 
lower  to  the  case  in  which  it  is  an  hyperbola.  The 
signs  in  (605  and  606)  are  derived  from  the  consid-  r  ^""') 
eration  that  when  y'  is  zero,  (p  is  equal  to  (p^  ;  but 
when  <p  is  zero,  we  have,  by  using  the  signs  as  be- 
fore, 

<?'  ±  To  =  0.  (608) 

87.  Corollary,    If  we  make  (fig.  4) 

t  =  PT,     t'  =  PT',  ^ 

yj  =3  the  inclination  of  P  T  to  the  axis  A  A'  \     (609) 
y  =:  the  inclination  of  P  T"  to  ^  A',  3 

The  inclination  of  Z.  i2  to  ^  ^'  is  </',  and  that  of  L'  R  is  <p'  : 
hence 

MN  z=z  L  R  cos.  (p  =:  A  cos.  (p  tan.  J  ('p  —  (p')    ^     /pinx 
M'  N=L'R  COS.  9'  =  A  cos.  <?'  tan.  ^  (rp  —  ^')    5     ^       ^ 
12 


134  INTEGRAL    CALCULUS.  [b.  V.  CH.  VI. 

Elliptic  integral  of  the  second  order. 

t  — —  A  tan.  X{(p  —  if')  .  *^        (611) 

COS.  V^  "^   ^  ^      COS.  T//  ' 

M'  N  ^  ,    ,  ,,       COS.   <Y)'  ,n^cx^ 

t  = ■  =  A  tan.  ^  (t  — 'P') .  ^ rr  .      (612) 

COS.  ^Z  "^   ^  '     COS.  V^'  ' 

But  the  construction  of  TP  gives 

»  JB 

tan.  V^=  -i  tan.  (/),    tan.  y  r=  —  tan.  H>'  ;  (613) 

whence 

_l^=V(l+tan.=  V')=V(l+5-ta,i.=  .) 

irr\/(l+tan.2  (f — g2  tan.2  (/))=\/(sec.2y — gS  tan. 2  ^ 
=::sec.  9'  V  (1— e^  sin.2  T)  z=  sec.  9  ^  .  <^,  (614) 
z=  ^  sec.  (p  cos.  a. 


In  the  same  way, 

1  J  tp'  cos.  &' 


(615) 


COS.  '^j'        cos.  y'        cos.  9' 
which,  substituted  in  (611  and  612),  give 
t  =A  tan.  J  (f — 9')  ^  g'  —  =F  ^  tan.  J  (^ — 9)')  cos.  q    (616) 
<'=:^  tan.  I  (<r— 9^')  ^  <?'=       -4  tan.  ^  (9—9')  cos.  fi'.  (617) 

88.  Corollary.    When  (/  is  zero,  (616  and  617)  become 

^0  =  i  tan.  ^  -Po  COS.  ^0  (618) 

fj  =  A  tan.  ^  ^o-  (619) 

89.  Corollary.  If  the  semicircle  A  R  A'  with  its  tangents 
R  L  or  R  L'  were  turned  round  ^  A'  as  an  axis,  so  as  to  be 
brought  above  the  plane  of  the  ellipse,  until  the  angle  which 
the  two  planes  made  with  each  other  were  one,  whose  cosine 


<5(  90.]  RECTIFICATION    OF    CURVES.  135 


Elliptic  integral  of  the  second  order. 


was  equal  to  the  quotient  of  B  divided  by  A,  the  ellipse  and 
its  tangents  would  evidently  be  the  projections  of  the  circle 
and  its  tangents.  The  angles  which  t  and  t'  made  with  the 
tangents  to  the  circle  of  which  they  were  the  projections, 
would  evidently  from  (616  and  617)  be  ^  and  &'.  A  consid- 
eration of  the  spherical  right  triangles  formed  at  the  points  of 
meeting  S  and  S'  of  the  tangents,  would  lead  anew  to  the 
same  equations  which  we  have  already  obtained. 

90.  Corollary.    The  equations  (616  and  617)  give,  by  using 
the  signs  as  in  (607), 

t'  :±:  tz=A  tan.  ^  (9—9')  (cos.  &'  —  cos.  6) 

=2  A  tan.  J-  (9— g)')  sin.  ^  (^'-f  5)  sin.  ^  (&—6').  (620) 

But  by  (350)  of  Trigonometry,  and  fig.  5, 

sin.  ^  (&'+&)  :  sin.  ^  {&—6')=ian.  |  9o  :  tan.  ^  (9— (Jp'),  (621) 
or   sin.  ^  {&'+&)  tan.  ^  {(p--(p')=s\n,  ^  (6—6')  tan.  ^  (p^  ;  (622) 

which,  substituted  in  (620),  gives 

t'  ^t  =z2  A  tan.  }  (o^  sin.2  ^  (&—&') 

=  A  tan.  2^  ^0  [1— cos.  (6—6')].  (623) 

In  the  same  way, 

to  it  #0  =  ^  tan.  \cfo{^  +  COS.  ^o)-  (624) 

Hence 
ti-t'^  (to—t)  =  A  tan.  i  (To  [cos.  (6-3')  +  cos.  6  J.  (625) 

But  by  (319)  of  Trigonometry, 

cos.  &Q  =:  —  cos.  (6  — 6')  -\-  2  sin.  6  sin.  6'  cos.^  J  (jp^  ,  (626) 


136  INTEGRAL    CALCULUS.  [b.  V.  CH.  VI. 

Sum  of  elliptic  integrals  of  the  second  order. 

< — ^ 

which,  substituted  in  (625),  gives  by  means  of  (604), 

t^ — /'±  (^0 — 0  =  2-4  tan.  h  (f^  cos.2  1  cp^  sin.  6  sin.  e' 
z=  A  sin.  cpQ  sin.  &  sin.  ^' 
=.  A  e^  sin.  cp^  sin.  q)  sin.  (p'.  (627) 

91.  Corollary.     If  Tq  (fig.  1)  is  the  point  for  which  (p  be- 
comes qpQ,  we  have 

A  E(p  =    TB=:s,  '] 

A  E  w'  —  T'  B—  s'  I 

^-^-  V         (628) 

AEcp^z=z  QB  —  s^.  J 

When  the  point  P  at  which  the  tangents  meet  are  situated 
upon  the  secondary  ellipse,  we  have 

TT'  zzzs  —  s'  \  (629) 

and  because  the  excess  of  the  sum  of  the  tangents  over  the 
included  arc  is  constant, 

or  s,-s  +  s'.=.t,+t;,-{t  +  t'y  (630) 

Hence,  by  (627  and  628), 

E(pQ-\-  E  cp' —  Eqi  =  e^  sin.  (p^  sin.  cp'  sin.  (f,     (631) 

m  which  g)^,  cp'  and  cp  are  subject  to  the  condition  (599), 
identical  with  one  of  the  following  conditions,  easily  deduced 
from  the  spherical  triangle  of  fig.  5,  by  means  of  (605)  and 
(606)  ; 

cos.  (p  r=  COS.  cpQ  cos.  <jd'  —  sin.  <Pq  sin.  cp'  j  gr,        (632) 

cos.  cp'  =  COS.  q)Q  COS.  cp  -[-  sin.  cp^  sin.  cp  j  cp',        (633) 
Bill  if  the  point  P  is  situated  upon  the  hyperbola,  we  have 
Q  T=zs—s, ,     Q  T'=s,-^s',     Q  T^=s^-^s, ;    (634) 


<5>   94.]  RECTIFICATION    OF    CURVES.  137 

Elliptic  integral  of  the  second  order. 

and  because  the  excess  of  the  difference  of  the  tangents  over 
the  difference  of  the  arcs  counted  from  Q  is  constant, 

t'  —  t-{-S'—2s^-\-s'=t'o  —  to  +  So^2s^, 

s+s'  —  s^z=to  —  t,  —  t'  +  t.  (635) 

Hence,  by  (627  and  628), 

E  (p  -\-  E  cp'  —  E  Vo  =  e^  sin.  (p  sin.  (p'  sin.  %  ,        (636) 

in  which  (p,  </  and  ^o  are  subject  to  the  condition  (599),  which 
is  identical  with  (633),  or  with  the  following  condition  ; 

COS.  (p  =  COS.  To  cos.  (p'  -\-  sin.  (p'  ^  9.  (637) 

92.  Corollary.  The  proposition  contained  in  (636)  is,  evi- 
dently, the  same  with  that  of  (631).  It  follows,  therefore,  that 
the  point  of  meeting  of  the  tangents  drawn  at  the  extremities 
of  the  arc  s^  —  s  of  (635)  is  upon  an  ellipse,  which  passes 
through  the  point  of  meeting  of  the  tangents  drawn  to  the 
extremities  of  the  arc  s',  and  which  has  the  same  foci  with  the 
given  ellipse  ;  the  same  may  be  inferred  with  regard  to  the 
tangents  drawn  through  the  extremities  of  the  arcs  Sq  —  s'  and 
5  of  (635).  It  follows,  in  the  same  way,  that  the  point  of 
meeting  of  the  tangents  drawn  at  the  extremities  of  the  arc 
5o  — s'  of  (630),  is  upon  the  hyperbola  which  passes  through 
the  point  of  meeting  of  the  tangents  drawn  at  the  extremities 
of  the  arc  s,  and  which  has  the  same  foci  with  the  given 
ellipse. 

93.  Corollary.  When  the  points  T  and  T'  coincide  at  the 
point  Q,  (636)  gives 

2  jE  y  J  :=  c2  sin.9  ,;,j  sin.  To  +  £  To .  (638) 

94.  Corollary.  The  supplements  of  the  angles  n — 5,  it — ^', 
n  —  ^0  of  the  triangle  of  fig.  5,  may  be  the  sides  of  a  spherical 

12* 


138  INTEGRAL    CALCULUS.  [b.  V.  CH.  VI. 

Elliptic  integral  of  first  order. 


triangle,  of  which,  re  —  (/),  it  —  9)',  n  —  (f^   are  the  opposite 

angles.     In  this  case,  since 

sin.  (n  — fl)  =  sin.  ^  and  cos.  (^r — ^)  =  —  cos.  fi,  &C.    (639) 

and 

1_ sin,  y  _  sin,  (p'   _  sin,  yp 

e        sin.  &         sin.  ^'         sin.  ^0     ' 
we  have,  by  putting 

^'J=zJ,(^^  .  A=^^l_-1  sin.sA,       (641) 

E'.^  =J*l  ^'.^;  (642) 

E'.&  4-  E'.&'  —■  E'.Bo^-^  sin.  &  sin.  6'  sin.  fl" 
'  e 

=  e  sin.  H>  sin.  g''  sin.  y.      (643) 
But  since  the  differentiation  of 

e  sin.  If  =:  sin.  & 

gives  ^    ^  ^  £Cos^  ^  ecos^ 

^  COS.  5  J  .  (p  ^         ' 

we  have  i^,  ^       f^  ai   a  —  T^    ^^e 

E'.  6  =/       ^'.    Q    ■=.   t       .  COS.  f/) 

We  have  also,  by  (581), 

COS.2  q)  r=  1  —  sin.2  y  =  1 ^  +  ^  (^  '/')2,     (647) 

which,  substituted  in  (646),  gives 

i  —  e-  /*(?      1       .1    Z*^ 


1  __e2  /»,^     1  1 


§   96.]  RECTIFICATION    OF    CURVES.  139 

Elliptic  integral  of  first  order. 

Similar  equations  may  be  found  for  E'  6'  and  E'  6^,  all  of 
which,  substituted  in  ((543),  give 

1—e^r     pp  J^     ,     /»r  J ,      /•To   J_"j 

e       Ly   0   ^'^     */    0    ^'f         J    0    ^'^J 

-\ —  {E.  (p-\-E  .  (f' — E  .  yj^ic  sin.  9  sin.  9^'  sin.  </>"  ;  (649) 
whence,  by  (636), 

/"^  J_  +  /""  _L    _     /•"»  J_  =  0.     (650) 

95.  Corollary.     The  integral 

is  the  elliptic  integral  of  the  first  kind,  and  is  denoted 
by  P  .  (p,  that  is, 

Hence,  by  (650), 

F  .  ./>  +  P.  ,/  —  F,n  =  0,  (653) 

where  9,  V'  and   9'o  are  subjected  to  the   same   conditions   as 
in  (636). 

96.  Corollary.  In  the  same  way  in  which  y  is  connected 
with  (f'  by  means  of  the  construction  of  fig.  1,  in  which  P  is 
upon  the  ellipse,  giving  by  {Qo^  and  631)  the  equation 

F.  T  =.  F.  <p'  +  JP.  yo ,  (654) 

other  points,  v",  ^'",  &lc.  might  be  found,  such  that 
Fcp'  -  F.  <p"  +  F.  (po 
F  cp"  z=z  F.  q>"'  +  F.(po,  &c  ;  (655) 


140  INTEGRAL  CALCULUS.      [b.  V.  CH.  VL 

Elliptic  integral  of  first  order. 

whence  F.<p   =^  F  cf"  -\-^  F.  cp^ 

z=Fcf"'-\-SF.(ro 

=  F.cp,  +  n  F.  cp, ;  (656) 

or  F.cf  —  Fcp.zzzniF.cp-^Fcp').  (657) 

97.  Corollary.  If  in  equation  (656)  (p^  vanishes,  (656)  be- 
comes F.  (p  =  71  .  F  (pQ  f  (618) 

and  it  is  obviously  easy  to  obtain  a  geometrical  construction  of 
the  corresponding  condition  between  cp  and  ^^  ,  by  taking 
several  successive  tangents,  BTP,  PT'P',  &,c.  as  in  (fig.  5), 
and  the  points  P,  P',  P",  &c.,  correspond  to  q),  cp',  cp",  &,c. 
The  tangents  might  also  be  drawn  to  the  circle,  from  the  suc- 
cessive points  P,  R',  R",  &c.,  of  the    ellipse,  of  which  the 

semiaxes  are  A'  and  .     A  similar  construction,  in  which 

B 

the  series  of  tangents  does  not  commence  with  JB,  would  sat- 
isfy the  conditions  of  (656  and  657.) 

98.  Corollary.  The  value  of  cp  for  the  arc  B  T  (fig.  4)  is 
equal  to  the  angle  which  the  corresponding  tangent  L  S  to 
the  circle  makes  with  the  transverse  axis  A  A'.  Denote 
by  X  the  angle  which  the  tangent  S  T  io  the  ellipse  makes 
with  the  conjugate  axis,  so  that 

^  =  J  TT  _  v'.  ■  (659) 

When  the  plane  of  the  circle  is  elevated  above  that  of  the 
ellipse,  the  spherical  right  triangle  formed  about  S  for  the 
centre  of  the  sphere,  has  cp  for  its  hypothenuse,  and  &  and  r  for 
its  legs.  It  is  represented  by  (fig.  6),  and  if  «  is  the  angle 
opposite  to  6f  and  r  the  angle  opposite  to  r,  we  have 


§  98.]  RECTIFICATION    OF    CURVES.  141 

Complete  integral.     Complementary  functions. 

COS.  a  rz:  —  =  ^(1 e~)   =  COS.  Q  Sin.  V  z=   COt.  (p  cot.  V'    (660) 

sin.  «  — e,  .  (661) 

COS.  xp  rz:  sin.  <p  sin.  v,  (662) 

COS.  fz=z  sin.  a  sin.  yj  =  e  sin.  yj.  (663) 

These  equations  give  by  the  differentiation  of  (660),  by 
representing  by  jPj  the  value  of  JT  corresponding  to  a  right 
angle,  and  observing  that  when  <p  is  zero,  r  is  also  zero,  and 
V  a  right  angle  ;  but  that  when  9  is  a  right  angle,  t  is  also  a 
right  angle,  and  V  is  zero, 

J  {e  .(p)  =  COS.  6  =  ^"  ,  ,  (664) 

COS.  «  sin.2  op 

Jjyj  .(p=  —  ,  (665) 

^  C0S.2  v^         '  ^        ' 

2 


2>i/;  .  <35 sin.  V  sin.2  go  sin.  go 1 

^(e.gp)  COS."  ^  cos.  V^  sin. 

1  1  1 


(666) 


^(1 — cos.^r)         \/(l-e"sin."^i/;)  ^(e.V) 

J     o^i^'f)  J    ^      ^(e.g') 

V^    -^(e-V')     t/     0     j{c.yj)      J    0   j{e.yj) 
=  F,—F{e.y^);  (667) 

SO  that  Fcp  and  -Fv^  are  two  functions  whose  sum  is 
the  function  Fi  which  is  called  the  complete  integral^ 
and  the  two  functions  are  called  complementary  with 
regard  to  each  other,  as  well  as  the  angles  w  andV',  upon 
which  they  depend. 


142  INTEGRAL    CALCULUS.  [b.   V.   OH.   VI. 

Transformation  of  elliptic  integrals. 

99.  Corollary.  The  three  angles  V^,  v^'  and  ^q^  which 
correspond  to  <r,  w'  and  (p^ ,  satisfy  the  equation,  equiva- 
lent to  (653), 

F.  ,p  +  F.y.'  —  F.  v^o  ^  0,  (668) 

when  they  are  subject  to  the  condition,  equivalent  to  (599), 

COS.  V'o  =^cos.V'  cos.  yp'  —  sin.  V'  sin.  V^'  ^  V^o  •       (669) 

100.  Corollary.  Denote  by  2  ;ir  the  angle,  which  FT 
(fig.  1)  makes  with  the  transverse  axis.  The  angle,  which 
F  T  makes  with  the  tangent,  is 

jTT  +  t/^  —  2z=:J^  —  (2;^  —  V^).  (670) 

The  projection  oi  F  T  upon  the  tangent  is,  therefore, 

i^TXsin.  (2;.-v),  (671) 

while  that  of  F'  T  upon  the  same  tangent  is 

F'  T  X  sin.  (2;^  — V);  (672) 

the  sum  of  which  is 

{FT  +  F'  T)  sin.  [2x  —  ^)  =  2  «  sin.  (2;^  — V').  (673) 

But  this  is  the  projection  of  FF'  =z  2  ae  upon  the  same  tan- 
gent, which  is  2«esin.  V^;  (674) 

and  therefore,  we  have  the  equation 

sin.  (2  j^  —  V^)  =  e  sin.  V^ ;  (675) 

whence,  by  (663), 

2x  —  ->p  =  ^7v-.v,  (676) 

The  equation  (675)  gives  by  putting 

'"  =  TT^'  (677) 

^."  cp=^  {e",  cp),  (678)' 

F':<p=F{e",cp);  (679) 


«5»   102.]  RECTIFICATION    OF    CURVES.  143 

Transformation  of  elliptic  integrals.  - 

~  (680) 

tan.  V=   ^'"-^^    ,  _L  =  A{e+iy-^es\n.''x)Je+\ y.x 

e-(~^os-  ^  X  '  COS.  If  e-\-  cos.  2  x  c-|-co3.  2  x  ' 

/^  X        14-ccos.  2  X 

J  yj  =  cos.  (2  X  —  V')  =  -^ .  cos.  i//,  (681) 

e  -\-  cos.  2  X  ' 

_,  2  cos.  (2  y  —  v^) 

Z>;^.^= ^^- ^ —  ,  (682) 

'^  cos.  (2  X — V)  -|-  e  COS.  V^  '  ^       ' 

i>;f.V_  2  2{e+\)j".x 


^  ip        COS.  (2  yir — V^)-f-c  COS.  v^       l-J-e'^-(-2e  cos.  2  x 
"■      (e  +  ir(y'.>rr     ~(e+l)/'.;^' 


(683) 


J  0   J^     J 


X  Dx  '^P  __ 

0     J  ■^J 


//  2  2  p" 

(  .  M^//-^-Jr. ^"- "^  =  4-  ^"- ^-  (6B4) 
0  (e+l)  ^"./      c4-  1  a/  « 

101.  Corollary.  If  ;r,  z', -^'q  correspond  to  V^,  V',  % ,  we 
have  the  equation 

i^."z  +  i^.";?'  — J^.";ro  =  0,  ,      (685) 

with  the  condition  that 

COS.  xq  =  cos.  X  COS.  X'  —  sin.  x  sin.  /  ^"  ;ko  .     (686) 

102.  Corollary.  In  the  same  way  in  which  F'  x  is  ob- 
tained from  F^^  another  function  F"x^  might  be  obtained 
from  F'  Xf  and  so  on,  until  a  series  was  obtained  in  which  the 
values  of  e,  e",  e"',  &,c.  form  a  series  of  eccentricities,  in  which 
each  differs  less  and  less  from  unity.  It  may  be  shown  that 
e"  differs  from  unity  less  than  e  does,  for  (677)  gives 

(689) 

1-^  ~(i+v^)'^(i+^)~(i+v^)-^(iW  '^' 


144  INTEGRAL    CALCULUS.  [b.  V.   CH.  VI. 

Multiplication  of  elliptic  integrals. 

in    which  the  factor  of    1 — e  is  evidently   less  than  unity, 
and  decreases  rapidly  with  the  decrease  of  1 — e.     The  value 
of  F.  (p  may  therefore   be  made  to   depend  upon   a  value  of 
F=  {e„.  cp„),   in   which   c„  differs  from  unity  by  as  small  a 
quantity  as  we  please,  and  we  have 

(690) 


103.  Corollary.  In  the  same  way  F  (ecp),  by  reversing  the 
above  process,  may  be  made  to  depend  upon  the  value  of 
F  (cn  (fn))  ^^  which  e„  is  as  small  as  we  please.  In  this  case 
e"  may  be  found  from  e  by  reversing  the  accents  in  (677)  and 
solving  the  equation  with  regard  to  c",  which  gives 

104.  Corollary.    If  we  put  in  (677) 

e  —  tan.2  ^  ?,  (692) 

we  have  e"  =  sin.  /5.  (693) 

105.  Corollary.    We  have  by  (681,  680,  683), 
COS.  2  X  = (694) 


J  -^  —  e  COS. 

yj 

(I        c2)cos. 

■  ^ 

'J  xp  —  e  cos. 

.^' 

J  rl>-\-  e  COS. 

Xfj 

c-f-cos.2;r=  rj-^^ ^— — ^^=r(^V^+ccos.  v)cos.'^  (695) 

J"  x^ 


1+6 


<§>   106.]  CIRCULAR    FUNCTIONS.  145 

Reduction  of  elliptic  integrals. 

/V^  ( J  yp-\-e  COS.  v^)2_    /» ^'  2(^T/;)^4-^  e  cos.  ^p  J  yj-(  l-e^) 

/^  /I -^  /•V  1 — e  Z*'/^  «  COS.  1// 

0  T+c  ""^  0  2"^  "^^  0      14-e 

=  ^-ni-e)F^+'-f^,  (696) 

which  may  serve  to  deduce  the  value  o^  E  {e  .  (f)  from  those 
of  E.(e„cp„),  in  which  c„  is  very  small,  or  differs  but  little 
from  unity. 

106.  Corollary.  Potential  functions  may  be  applied  to  the 
hyperbola  very  nearly  in  the  same  way  in  which  circular  func- 
tions have  been  applied  to  the  ellipse.     Thus  if  we  put 

A  Cos.  (p  =  X,      B  Sin.  (p  =  y,  (697) 

X  and  y  are  the  coordinates  of  the  hyperbola,  of  which  the 
equation  is 

(i)-(fy->-       («^«) 

The  length  of  the  hyperbolic  arc  is,  by  putting 

^  =  v(l+2),  (699) 

s=Bfs/  (l  +  c^  Sin.s^)). 

If  we  let 

^  (c  (t)  =  \/  (1  +  e^  Sin.'-J  q)  (700) 

ar  {ecf)=    /'^^(eT),  (701) 

»/  0 

we  have  s  ^=  B  ;j  {c(f).  ("62) 

107.  Corollary.  The  condition  that  the  point  of  contact  in 
(fig.  2)  is  upon  another  hyperbola  which  has  the  same  foci  with 

13 


146  INTEGRAL    CALCULUS.  [b.  V.   CH.  VI. 

Hyperbolic  integrals. 

the  given  hyperbola,  is  expressed  algebraically  by  the  equa- 
tion 

Cos.  </)q  :=  Cos.  (f>  Cos.  cp' —  Sin.  cp  Sin.  (f'^  r  ^o  t    (703) 

and  corresponds  to  the  equation 

3"  (f'+  3"  <?'  —  a-  <JPo  ==  c~  Sin.  (jDq  Sin.  cp  Sin.  cp',      (704) 

in  which  cp',  qi^ ,  cp,  correspond  respectively  to  the  last  and  first 
points  of  the  hyperbolic  arch  included  by  the  two  tangents,  and 
to  the  last  point  of  the  arc  of  which  the  first  point  is  the  ver- 
tex. 

108.  Corollary.    In  the  same  way,  if  we  put 
we   have,  with  the  condition  (703),  the  equation 

^  V'  +dr  V  —d  n  =  0.  (706) 

109.  Corollary.  When  e  was  changed  into  its  reciprocal 
in  §94,  it  became  greater  than  unity,  and  ceased,  therefore, 
to  correspond  to  an  ellipse.  It  may  easily  be  shown  that,  in 
this  case,  however,  the  transverse  axis  and  the  angle  cp  become 
imaginary  of  the  form  A  \/ — 1  and  cp  s/ — 1 ;  so  that  the  ellipse 
changes  into  the  hyperbola,  of  which  B  is  the  transverse  axis, 
and  A  the  conjugate  axis ;  and  the  circular  change  into  po- 
tential functions.  This  case  is,  therefore,  the  same  as  the  one 
just  investigated,  and  it  may  be  remarked,  that  the  equilateral 
hyperbola  takes  the  place  of  the  circle. 

110.  Corollary.    If  ^  is  determined  by  the  condition  that 

cos.  a  =  Tan.  cp,  (707) 


<§>   111.]  RECTIFICATION    OF    CURVES.  147 


Length  of  hyperbolic  arch. 


we  have    sin.  ^  =: ,     cot.  ^  =z  Sin.  <jp;  (708) 

Cos.  (jp 

whence  Dq.(p  =  —  cosec.  ^ 

e  jj(\  —pr-  sin.2  5) 

F.^^^s/  (1+^2  cot.^O-   ^     3i„;a (^^^) 

=  e  cosec.  5  ^.  (e'".  5),  (710) 

if  ,-^v'-^^.  (711) 

Hence 

•9?        1  ph.^  D&^ 


j:  '(p  = 


6       r  (p 


o'   r  (p  J 


111.   Corollary.    Let  w  be  so  taken  that 

Sin.  <p  =  -  tan.  w ;  (713) 

e 

and  we  have,  by  putting 

e'  =  V(l-^,),  (714) 

_.  /v/  (1 — e'2  sin. 2  w)  ^  (e'  w) 

Cos.  (f  ■=:  — ■ 


cos. 

0) 

COS. 

(0 

A'  w 

(715) 

cos.  CO  ' 

1 

(716) 

COS.  w    * 

1 

r  <f  -= 


Dio.^ .  /'71'7\ 

e  y  w  .  COS.  w  C*  1 ' ) 


148  INTEGRAL    CALCULUS.  [b.  V.   CH.  VI. 

Length  of  the  hyperbolic  arch. 

Hence  the  length  of  the  arc  of  the  hyperbola  is 

S=B  3;.(p:=Br^^    Dai  <P  .  F  .  (f 


B    /»«  1 


C -#   0        COS. 2  (a  J'  vi) 

_B    /*w     1  B    /^o,      sin. 2  t,j 

~  C  J   0       J'  oi  e   J  ^     C0S.2   OJ  ^'  w 

(718) 

:r:  _  P'  w — B  e  E'  o)-\-B eg      {   /  w-j- ^ -/ ) , 

e  •/  0  \  ^  *"  / 

But,  we  have, 

,  (l-e'2)tan.2cj      ^' c.     ,rl-c'2  -i 

.</'  w  e'2  sin.2  w 


C0S.2  CO  -i'  CO 

==2>.(tan.  CO  ^'(v);  (719) 

which,  substituted  in  (718),  gives 

s  =  -  F'.  oi  —  B  e  E'o^-\-Be  tan.  w  j'  « 

=  V(^^  +  ^^)r^^^  — ^'''^  +  tan.  CO  ./'wl.  (720) 


<§)   113.]  RECTIFICATION    OF    CURVES.  149 

Arc  of  curve  of  double  curvature. 

112.     Examples.* 

1.  Prove  that  the  tangent  let  fall  from  the  centre  of  the 
hyperbola  upon  the  tangent  is-in  the  notation  of  §  111. 

Be  tan.  to  ^'w  —  ^(^2_|_^2)  tan.  «  ^'<«  (721) 

2.  Prove  that  if  e"  and  x  are  so  taken  that 

sin.  {^x —  f")  =  e'  sin.  %  (723) 

the  length  of  the  arc  of  the  hyperbola  is 
5=rV(^'+^')[JS''"-2(l+e')^">^+2e'sin./+tan.c.  /J.  (724) 

113.  Definition.  A  curve  of  double  curvature  is  a 
curve  all  the  parts  of  which  do  not  lie  in  the  same 
plane. 

114.  Problem.  To  find  the  length  of  an  arc  of  a 
curve  of  double  curvature. 

Solution.  Since  the  length  of  an  infinitesimal  arc  of  the 
curve  is  equal  to  the  distance  apart  of  the  two  infinitely  near 
points  x,y,z  and  x-\-dx,  y-\-dy,  z-\-dz\  or,  algebrai- 
cally, 

d  s  —s/  {dx^  -\-dy^  -\-  dz^)  ',  (725) 

the  length  of  an  arc  is 

s  =f^(Dx2-{-Dy2+Dz2),  (726) 

*The  examples  will  not  hereafter  be  strictly  confined  to  the  subject 
of  the  chapter,  but  will  extend  to  any  exercises  suggested  by  the  in- 
vestigations in  the  chapter. 

13* 


150  INTEGRAL    CALCULUS.  [b.  V.   CH.  VI. 

Helix. 


115.    Examples. 

1.    To  find  the  lencrth  of  an  arc  of  the  helix. 

Solution.  The  helix  is  a  curve  formed  by  a  string  wrapped 
round  a  cylinder  in  such  a  way  as  to  make  a  constant  angle 
with  the  side  of  the  cylinder.  Hence,  if  the  plane  of  xt/  is 
that  of  the  base  of  the  cylinder,  if  the  centre  of  the  base  is 
the  oricrin  of  coordinates,  and  if 

Q  z=  radius  of  the  base  of  the  cylinder  "^  ^ 

(p  =:  the  angle  which  the  projection  of  radius  vec-    | 

tor^on  the  plane  of  a?y  makes  with  the    !     fjcy^x 

axis  of  X, 
a  :=.  the  angle  which  the  string  makes  with  the 

side  of  the  cylinder, 

the  equations  of  the  helix  are 

x^=:q  cos.  9',     y  =  ?  sin.  qp,     2  =  ^  <r  cot.  o.        (728) 

Hence 


Q  ^ 


sm.  « 


(729) 


2.  To  find  the  length  of  the  arc  of  the  curve  formed  by 
winding  a  string  round  a  solid  of  revolution,  in  such  a  way  as 
to  make  a  constant  angle  with  the  meridional  arc  of  the  sur- 
face. The  meridional  arc  of  a  surface  of  revolution  is  the 
intersection  of  the  surface  with  a  plane  passing  through  the 
axis  of  revolution. 


«§>    115.]  RECTIFICATION    OF    CURVES.         '  151 

Rhumb  line. 

Ans.  With  the  notation  of  the  preceding  example,  the 
length  of  the  arc  is 

s=z  sec  «/,  .  V  [l+(^.  0^]  =  s'sec.  «;  (730) 
where  s'  =  the  arc  of  the  meridional  section,  corresponding 
to  the  required  arc  s. 

The  angle  ^  may  be  substituted  for  z  in  (730)  by  means  of 

the  equation 

/Z>.  s' 
— - —  •  .("3i) 

3.  To  find  the  length  of  the  arc  in  Example  2,  when  the 
solid  is  a  right  cone. 

Ans.    If  (?  =1  the  angle  which  the  side  of  the  cone   makes 
with  the  axis, 
Zq  =  the  value  of  z  at  the  beginning  of  the  arc 
s  =:  (z  —  Zq)  sec.  «  sec.  ^ 

if  sin.  ^  cot.  a 
=  sec.  a  sec.  ^  .  e.  ("^32) 

4.  To  find  the  length  of  the  arc  in  Example  2,  when  the 
solid  is  a  sphere  j  that  is,  to  find  the  length  of  an  arc  of  a 
rhumb  line. 

Ans.    If  J?  =  the  radius  of  the  sphere  "^ 

6  =  the  inclination  of  the   radius  vector   I 

to  the  axis  of  z,  r    C'^S) 

6q  =  ink  value  of  ^  at  the  beginning  of  the   i 
arc, 

the  length  of  the  arc  is 

s  =  R{&  —  &,)sec.  a  (734) 

and  (f  may  be  substituted  for  ^  by  means  of  the  equation 

(p  =  log.  tan.  J  ^  —  log.  tan.  J  6^  .  (735) 


152  INTEGRAL  CALCULUS.       [b.  V.  CH.  VI. 

Shortest  line. 

5.  To  find  the  length  of  the  arc  in  Example  2,  when  the 
solid  is  an  ellipsoid  of  revolution.  ^ 

Ans.    I^  A  =  the  semi-transverse  axis,       ^ 

e  =  the  excentricity,  \    (736) 

6        corresponds  to  <p  in  (574),     ) 
the  length  of  the  arc  is 

5  ==  A  sec.  a{E&  —  E  ^o),  (737) 

and  (p  may  be  substituted  for  ^  by  means  of  the  equation 

,  =  log.  (■y-c°s.^)sin.^o  ^ '"'";+ ^^  ,  (738) 

in  the  case  of  the  oblate  ellipsoid.      But  in  case  of  the  prolate 
ellipsoid,  the  equation  is 

J  &  -L.  ^  n  —.  e^)  sin.  &      ,    ,        COS.  ^o 

<p  =  log-  T.        /M — riVT^r-:-  +  ^^g- 


^^o  +  /v/(l — e^)  sin.  ^0  cos.  ^ 

+ ^ tang.-^^ ^in.  ^  -  sin.  M 

^^(l_e2)        S       l-sin.fi-sin.^o+2sin.  asin.ao  ^ 

116.  Problem.  To  find  the  shortest  line,  which  can 
be  drawn,  subject  to  giveii  conditions. 

Solution.  The  variation  in  the  length  of  the  curve  arising 
from  any  very  small  change  in  its  equati(|j;i,  which  may  be 
made,  consistently  with  its  conditions,  is  in  general  propor- 
tional to  the  magnitude  in  the  change  of  the  equation ;  it 
must,  therefore,  change  its  sign  with  the  change  of  sign  in  the 
variation  of  the  equation.  Thus  if  ^  z  denote  the  variation  of 
the  equation,  we  shall  have  for  the  variation  in  the  length  of 
the  curve, 


•J    116.]  RECTIFICATION    OF    CURVES.  153 

Shortest  line. 


ds  =  D,s  .Sx  +  ^D^^s.Sx^  +  Slc,  (740) 

the  second  member  of  wliich  is  reduced  to  its  first  term  when 
5  a;  is  an  infinitesimal.  But,  in  the  case  of  a  maximum  or 
minimum,  S  s  should  not  change  its  sign  with  <^x,  and  therefore 
this  first  term  should  be  wanting,  that  is,  the  variation  of  5 
should  be  zero,  while  the  second  variation  of  s  should  not  van- 
ish ;  or  if  the  second  variation  should  chance  to  vanish  with 
the  first,  the  third  variation  must  also  vanish.  The  principles 
of  finding  this  class  of  maxima  or  minima  are  the  same  with 
those  of  B.  II.  Chap.  VIII.  But  the  process  of  finding  a 
maximum  or  minimum  of  a  definite  integral,  such  as 

dependent  upon  a  variable  function,  is  quite  different  from  that  for 
the  ordinary  maximum  or  minimum  ;  and  such  problems  are 
often  considered  by  themselves,  under  the  title  of  the  Mdhod 
of  Variations. 

The  function  (741)  can  vary  in  two  ways,  either  by  the 
variation  of  the  functions  upon  which  s  depends,  or  by  the 
variation  of  the  limits  of  the  integration.  The  condition  that 
it  is  a  maximum,  is,  therefore,  expressed  by  the  equation 

(Jpi   Ds=zss,—Ss^-\-  r^^  SDs-0.  (742) 

The  value  of  Z>  s  usually  depends  upon  many  variables, 
t,  X,  1/,  &LC.,  and  their  differential  coefficients,  in  such  a  way 
that,  if  t  is  taken  for  the  independent  variable,  any  change  in 
the  functions  by  which  a-,  y,  &,c.  depend  upon  t,  gives  the 
equations 

ds,=Ds,.dt,,     Ss^z=:D  So  .  ^t^,  (743) 

d  Dsz=X3x+YSi/+&Lc.   -\-X'S  Dx-\~Y'3Dij-{-&.c. 

+X"  i  2>2  a;_|_  Y"  $  Z>2  3,-|-&,c.  ;  (744) 


154  INTEGRAL    CALCULUS.  [b.  V.   CH.  VI, 

Method  of  Variations. 

which,  substituted  in  (742),  give 

-\- X"  8  D"^  xJ^&LC.    +F,Jy+&c.)=0.  (745) 

But,  by  (262), 

/.  X'  d  D  x=f.  X'  D  5  x=X'  d  x—fDX'  .  $x        (746) 
/.  X"  d  D^xz=f.  X"  Z/2  8xz=iX"  D  8  X—fDX".  D  $  x 

—X'  D  d  x-D  X".  d  x+f  i>2  X".  S  X,  &LC.  (747) 

The  terms  in  the  last  members  of  (746  and  747),  which  are 
not  under  the  sign  of  integration,  must,  in  passing  to  the  defi- 
nite integrals,  be  referred  to  the  limits  of  integration  But  it 
must  be  observed,  that  the  variations  in  (746  and  747)  are 
taken  upon  the  supposition  that  the  independent  variable  t 
does  not  itself  vary,  and  that  only  the  functions  vary,  by  which 
Xj  7/,  fcc.  are  connected  with  it ;  whereas  the  limits  of  inte* 
gration  may  themselves  necessarily  vary  with  a  change  of  this 
function,  and  therefore  t^  and  t^  are  supposed  to  vary.  If, 
then,  8' Xq  ,  ^'^Q,  iS^c.  denote  the  variations  arising  from 
the  change  of  the  functions,  the  values  of  the  complete  varia- 
tions are 

3x=zd'x^-\-  Dx^.St^,  6lc.  (748) 

whence  8'  x^z=z  ^  x^  —  D  x^  J  t^,  &c.  (749) 

Hence  (746  and  747)  give 

f^^^  XdD  x—X[  3'x^—X'o  -5'  ^o—fl^  DX.dx       (750) 


^^X'5D^xz=X[  Dd'x^—DX';  d'x^—X'^DS'x^ 


f 

—  DX;  6'x^+J^^^^  D^X".  <5  X,  &,c.  (751) 
in  which  S'  is  given  by  (749). 


§   116.]  RECTIFICATION    OF    CURVES.  155 

Method  of  Variations. 

These  equations,  substituted  in  (745),  give 

+(X;-  &c.)  D  S'.  x^—iX'^-SLc)  D  ^'  z^-f  &,c. 

(752) 
The  terms  of  (752),  which  are  under  the  sign  of  integration, 
express  a  variation  which  belongs  to  each  point  of  the  curve 
independently  of  all  the  other  points,  and  which  must,  there- 
fore, be  equal  to  zero  for  each  point ;  which  gives  the  general 
equation 

{X—D  X'-\-D^X"—&LQ,)  S  X+&C.  =  0.         (753) 

The  variables,  t,  x,  y,  &c.  may  be  bound  together  by  some 
conditions,  represented  by  the  equations 

i  =  0,     M=:0,  (754) 

in  which  L,  M,  may  be  functions  of  t,  x,  y,  &,c.  The  varia- 
tions of  these  equations  will  then  give  linear  equations  between 
^Xj  ^y,  &:,c.  from  which  the  values  of  some  of  the  variations 
iXj  <5y,  &c.  can  be  determined  in  terms  of  the  others.  These 
values,  substituted  in  (753),  will  reduce  the  number  of  varia- 
tions in  (753)  to  the  smallest  possible  number,  and  those  which 
remain  will  be  wholly  independent  of  each  other,  and  there- 
fore their  coefficients  must  vanish.  The  equations,  thus  ob- 
tained from  making  these  coefficients  equal  to  zero,  will  be 
the  required  equations  of  the  shortest  time. 

If,  in  addition  to  the  equations  (754),  the  limits  of  the  curve 
are  subject  to  peculiar  conditions ;  these  conditions,  with  those 
of  (754),  referred  to  the  limits  of  the  curve,  may  be  combined 
with  the  terms  of  (752),  which  are  not  under  the  sign  of  in- 
tegration, and  the  equations  for  determining  the  extreme  points 


/ 


156  INTEGRAL    CALCULUS.  [b.   V.    CH.  VI. 

Maximum  or  minimum  of  definite  integrals. 

of  the  curve  may  be  found  by  the  same  method  by  which  the 
equations  of  the  curve  itself  are  found. 

117.  Corollary.  The  preceding  process  for  finding 
the  minimum  of  (741),  may  be  apphed  to  finding  the 
maximum  or  minimum  of  any  definite  integral,  such 
as 

'■•  V,  (756) 

by  changing  in  the  various  formulae  D  s  into  F. 

118.  Corollary.  The  number  of  the  variations  ^x,  Sy,  &lc. 
determined  by  (754),  is  plainly  equal  to  the  number  of  the 
equations  of  (754).  The  number  of  the  variations  left  unde- 
termined, therefore,  in  (753),  and  consequently  the  number 
of  equations  obtained  from  (753),  is  equal  to  the  number  of 
the  variations  not  determined  by  (754).  The  whole  number 
of  equations  then  of  the  required  curve,  is  equal  to  the  whole 
number  of  the  variables  3",  y,  z,  &<c.,  among  which  the  inde- 
pendent variable  is  not  included;  that  is,  there  are  just  as 
many  equations  as  are  required  to  determine  the  curve. 

In  the  same  way,  it  may  be  shown  that  there  are  just 
enough  equations  to  determine  the  extreme  points  of  the 
curve. 

119.  Corollary.  The  following  method  of  eliminating  the 
variations  from  (753),  which  are  determined  by  (T54),  is 
more  symmetrical  than  the  usual  one,  which  is  proposed  in 
§  116.  Multiply  the  variation  of  each  of  the  equations  (754) 
by  some  quantity,  such  as  ^.,  u,  &bc.  and  add  the  sum  of  all 
the  products  to  (753).  The  values  of  /,  u,  &lc.  may  be  de- 
termined by  putting  equal  to  zero,  the  coefficients  of  just  as 


'Jj   120.]  RECTIFICATION    OP    CURVES.  157 

Method  of  variations. 

many  of  the  variations  ^  x,  si/,  &,c.  Tiie  substitution  of  these 
values  of  x,  n,  &:,c.  in  the  other  coefficients,  will  reduce  (753) 
to  an  equation,  from  which  as  many  of  the  variations  have  dis- 
appeared as  there  are  equations  (754).  The  remaining  co- 
efficients, being  those  of  independent  variations,  must  therefore 
be  equal  to  zero ;  that  is,  each  of  the  coefficients  in  the  sum 
formed  by  the  addition  of  the  products  of  (754)  by  x,  ii,  &lc>. 
to  (753),  may  be  put  equal  to  zero,  and  x,  ,«,  Sfc.  may  be  elim- 
inated from  the  result  by  the  usual  process. 

120.  Corollary.  If  all  the  variables  had  been,  in  the  outset, 
eliminated  from  Ds  (741)  or  F  (75G),  which  could  have  been 
eliminated  by  means  of  the  equations  (754),  the  remaining 
ones  would  have  been  independent  of  each  other,  and  would 
have  given,  at  once,  from  (753), 


X—D  X  -\-  Z>2 X"  —  &LC.  =  0, 
Y—  DY'  +  2>-^  Y"  —  &,c.  =  0,    &.C. 


}     (757) 


If,  moreover,  certain  of  the  variables,  and  among  them  the 
independent  variable,  had  been  taken  so  as  to  be  the  very 
functions  of  the  variables  which  were  constant  under  the  ad- 
ditional conditions  at  one  of  the  limits,  as  that  of  ^q  ;  we 
should  have  for  those  variables 

8t^={),  &.C.  (758) 

and  there  would  have  been  no  additional  conditions  between 
^  ^0  »  ^Voi  ^^-y  which  in  this  case  would  not  differ  from 
•5'  Xq  ,  5' yQ  ,  fee.  ;  so  that  (752)  would  give 

X',—D  X;'4-&c.=0,     Y',—D  F'J-f  &c.=0,  &,c.     (759) 
Xq— &c.  =  0  l"o— ^G-  —  0,  <S6C.  (7G0) 

of  which  (759)  are  the  same  with  (757)  referred  to  this  ex- 
14 


158  INTEGRAL    CALCULUS.  [b.   V.   CH.  VI. 

Shortest  line  upon  a  surface. 

tremity.  The  equations  (758),  however,  involve  the  hypothe- 
sis that  there  is  no  condition,  by  which  both  extremities  are 
bound  toiiether. 


121.   Corollary.    If  the  curve  is  referred  to  rectangular  co- 
ordinates X,  y,  2,  of  which  X  is  the  independent  variable,  we 

have 

Ds=^  {\+D  y^+D  ^2),  (761) 

^Ds^p^^Dy+^^Dz;  (762) 

whence  y—^      Z   —^—  ) 

~  Ds  '         ~  Ds  '  V     (763) 

Y—0,    Y"-0,&LC.  J 

These  equations,  substituted  in  (753  and  752),  give 


'Ds,ix-Ds,Sx,  +  ^i-y^  + 


Dz 


§f°''^o-^^'%  =  0.  (765) 


122.  Corollary.  Any  condition  between  the  rectangular 
coordinates  of  the  preceding  article  must  be  expressed  by  an 
algebraical  equation,  which  may  be  regarded  as  the  equation 
of  a  surface  iqjon  which  the  shortest  possible  line  is  to  he 
drawn. 

If  the  positions  of  the  axes  of  coordinates  are  so  taken  that 
one  of  the  axes,  that  of  z  for  instance,  is  perpendicular  to  the 


<J    123.]  RECTIFICATION    OF    CURVES.  159 

Line  of  least  curvature. 

surface   at  one  of  the  points  x,  y,  2,  through  which  the  curve 
passes,  we  should  have,  at  this  point, 

Sz=zO,  (766) 

and,  therefore,  by  (761), 

n(p^)=0.  (767) 

But  the  plane  of  the  axes  of  x  and  1/  is,  at  this  point,  parallel 
to  the  tangent  plane,  and 

Ds 

is  the  cosine  of  the  inclination  of  the  curve  to  the  axis  of  r/  ; 
so  that  by  (767)   this   inclination   is   constant. 

Hence  the  direction  of  the  shortest  line  drawji  vpon 
a  surface^  has  at  each  point  no  curvature  in  the  direction 
of  the  tangent  plane  ;  it  has,  then,  less  curvature  at 
each  point  than  any  other  curve  drawn  upon  the  same 
surface  through  that  point,  and  having  the  same  tan- 
gent with  it ;  that  is,  it  has  the  tnaximinn  radius  of 
curvature  of  all  lines  which  have  a  cominon  tangent^ 
and  are  drawn  upon  the  same  surface  ;  it  coincides,  then, 
with  the  direction  of  a  riband  which  is  woujid  round 
the  surface  in  such  a  way  as  to  bend  only  towards  the 
surface,  without  bending  in  the  tangent  plane  either  to 
the  right  or  left. 

123.  Corollary.  Upon  any  surface  whatever,  such 
as  a  cylinder  or  cone,  formed  by  the  bending  of  a  plane, 
and  which  is  designated  as  a  developable  surface^  the 
shortest  line  becomes  a  straiglit  line  icJien  the  surface 
is  bent  back  into  a  plane ;  and  it  may  be  remarked,  that 


160  INTEGRAL    CALCULUS.  [b.   V.    CH.  VI. 

Developable  surface.  Geodetical  line. 

any  surface  formed  by  the  motion  of  a  straight  line, 
which  remains  in  two  successive  positions  in  the  same 
plane,  is  developable. 

124.  Corollary.  It  is  obvious,  from  (353  of  vol.  1), 
that  the  hyperboloid  of  equation  (350  of  vol.  1)  is  not  de- 
velopable, and  that  therefore  the  preceding  corollary  is 
not  applicable  to  it,  although  it  may  be  generated  by 
the  motion  of  a  straight  line. 

125.  Corollary.  The  shortest  curve  upon  the  sphere 
is  an  arc  of  a  great  circle. 

126.  Corollary.  The  curve  drawn  upon  the  surface 
of  the  earth,  u])on  the  principles  of  *§^  122,  is  called  the 
geodetical  curve^  and,  therefore,  this  is  the  shortest  curve 
which  can  be  drawn  upon  the  earth^s  surface. 

127.  Corollary.    If  the   position  of  the  axes  is  taken  as  in 

^  122,  but  with  the  condition  that  the   axis  of  x  shall   be  the 

normal  to  the  surface,  we  have  for  the  point  a^o  '  i^'o  »  ^o  » 

^^0=0;  (768) 

whence,  by  (765), 

Dy,5y,  +  Dz,5z,^(i,  (769) 

Since  the  point  a;^^  ,  y^  ,  z^  ,  is  upon  the  given  surface,  any 
additional  condition  for  this  point  would  be  equivalent  to  re- 
quiring it  to  be  upon  some  other  surface,  so  that  it  would  have 
to  be  at  the  common  intersection  of  the  two  surfaces.  The 
first  member  of  (770)  expresses,  then,  the  tangent  of  the 
ancrle  which  this  intersection  makes  with  the  axis  of  z,  while 
the  second  member  expresses  the  negative  of  the  cotangent  of 
the  angle  which  the  required  curve  makes  with  the  same  axis. 


«§)    128.]  RECTIFICATION    OF     CURVES.  161 

Shortest  line  upon  a  surface. 


The  shortest  curve  which  can  be  drawn  upon  a  given 
surface  from  one  curve  upon  that  surface  to  another 
curve,  or  to  a  point  upon  the  surface,  is  perpendicular 
at  either  extremity  to  the  limiting  curve  at  that  ex- 
tremity. 

128.  Corollary.  If  ihe  required  line  is  subject  to  no  con- 
dition except  at  its  extremities,  the  variations  in  (764)  are 
entirely  independent,  which  gives  tiie  equations 

that  is,  since  the  direction  of  the  axes  is  wholly  arbitrary,  the 
required  line  has  no  curvature  in  any  direction,  and  is^  conse- 
quently, a  straight  line. 

If  the  extremity  x^  ,  ?/q  ,  Zq  ,  is  subject  to  a  condition,  that 

it  must  be  upon  a  given    surface  ;   the   norma!   to  that  surface 

at  the  extremity  of  the  line  may  be  taken  for  the  axis  of  Xq  , 

which  gives 

sx^:^.{),  (772) 

and  leaves  ^y^  and  ^z^  arbitrary  ;  whence,  by  (765), 

^=  0,         ^-^  =  0 ;  (773) 

D  So  IJ  Sq 

that  is,  the  cosine  of  the  angle  which  the  required  line  makes 
with  the  surface  in  each  direction  is  zero  ;  or,  in  other  words, 
the  required  line  is  perpendicular  to  the  surface. 

If  the  extremity  z„ ,  y^ ,  Zq  >  is  subject  to  two  conditions, 
that  is,  if  it  is  at  the  intersection  of  two  given  surfaces ;  let 
this  line  be  the  axis  of  Zq  ,  and  we  have 

^Xoz=0,     ^yo  =  0,  (774) 

14* 


162  INTEGRAL    CALCULUS.  [b.   V.   CH.   VI. 

Shortest  line  between  two  surfaces  or  two  lines. 

whence  (765)  gives 

fe  =  0,  (775) 

or  the  required  line  is  perpendicular  to  the  given  line.    Hence 

The  shortest  line  which  can  be  drawn  between  two 
given  surfaces,  or  two  given  lines,  or  a  line  and  a  sur- 
face, or  a  point  and  a  surface,  or  a  point  and  a  line,  is 
the  straight  line  which  is  perpendicular  to  the  surface 
or  line  at  the  corresponding  extremity. 

129.  Corollary.  If  the  shortest  line  is  required  to  be  drawn 
upon  a  surface  of  revolution,  let  the  axis  of  z  be  the  axis  of 
revolution,  let  u  be  the  projection  of  the  radius  vector  upon  the 
plane  of  x  y,  and  let  (p  be  the  angle  which  u  makes  with  the 
axis  of  oc ;  and  we  have,  by  taking  z  for  the  independent  varia- 
ble, 

Ds  =  ^  {u^  D  (/'2  ^  Bu^+l),  (776) 

But  by  the  equation  of  the  surface,  w  is  a  given  function  of 
2,  and,  therefore,  not  subject  to  variation.     Hence 

3Ds=  .^        ^ .  (777) 

The  equation  gives,  then, 

D  --^  =  0,  (778) 

U  s 

the  integral  of  which  is 

-jf^  =  C;  (779) 

J  J  s 

in  which  C  is  an  arbitrary  constant,  and  the  independent  va- 
riable may  be  any  variable  whatever,  because  it  is  only  the 
ratio  of  two  differential  coefficients  which  enters  into  (779). 


<5>    130.]  RECTIFICATION    OF    CURVES.  163 


Geodetic  curve  upon  the  oblate  ellipsoid. 


130.    Examples. 

1.  To  find  the  shortest  line  which  can  be  drawn  upon  the 
oblate  ellipsoid  of  revolution. 

Solution.  Let  A  be  ihe  greater,  and  B  the  smaller  semi- 
axis  of  the  generating  ellipse,  and  c  the  eccentricity  ;  we  have 
for  the  equations  of  the  ellipse,  as  in  (575  and  576), 

X  ^1  A  sin.  ^,     y  =B  cos.  &  ;  (780) 

and  X  in  this  equation  is  the  same  with  u  in  (779),  and  y  is 
the  same  with  z.  Hence  (776  and  779)  give,  by  taking  ^  for 
the  independent  variable,  and  using  the  notation  of  elliptic  in- 
tegrals, 

D  s^=A^  sin. 2  &  D  f^+A^  cos.2  &-\.B2  sin.2  & 

z=iA2  (sin.2  6  D  <r2+^a2)_.41^^  D  <f2,  (781) 

'^~  sin.  a  V  (^^  sin.2  ^_C2)  *  ('^^) 

Let  «,  T^,  and  e  be  taken,  so  that 

C^r^sin.  a,  (783) 

cos.  V^  ■=!  COS.  5  sec.  «,  (784) 

, e  COS.  « 

^  -V(l-«^sin.^'  ^^^^^ 

and  we  have,  by  taking  t^  for  the  independent  variable, 

sin.  ^  Z>,^  ^  z=i  COS.  «  sin.  V',  (786) 

^^2— -i_g2  (l_cos.2  a  COS. 2  v^)zz=l — e2(sin.2  a-j-cos.2  a  sin.2  1//) 

=(1— c2  sin.2  u)  (l_e'2  sin.2  ^j 

—  (I_e2  sin.2  a)  ^'  V'2^  (737) 


164  INTEGRAL    CALCULUS.  [b.   V.   CH.  VI. 

Geodetic  curve  upon  the  ellipsoid.     Elliptic  integral  of  the  third  order* 

V  (^-  sin. 2  <3— C2)  =  yl  V  (sin.2  ^— sin.2  «) 
z=zA  \/  (cos.2  Li — COS. 2  &)  ^^  A  COS.  «  sin.  V',  (788) 

^  _  C^^  2)./.^ sin.  «  V  (l-<^^  sin.2  g) .  ^^  y/ 

'^  '^~sin.  fi\/(^'sin.2  6-C2j—  sm.2  d 

__   v/(l—e^  sin.2  «)  ^>  _  V(I-c- sin.2  a)  y  ,/;2 

sin.«(l+col.2asin.2  t/^)        sin.  "  (l-j-cot.2  «  g^,^  2  ^^j  ^/ 1/; 

(789) 

sin.  «  r~      1-f-e^cos.  2  a  €^cos.^«~l 

"~cos.2  «  yy/( J -e^  sin.'-^  u)  L (i  +  cot.''' «  sTii^''  V')  ^' -^         ^' V'  J' 

Hence,  if  we  adopt  the  notation 

n(n.erp)  =1  f^  , }^ ,  (790) 

J    0  (1+/1  sin.2  v^)  ^^  '  '^        / 

and  put  n  =  cot. 2  «,  (^91) 

(789)  gives  ^-^^   ^ 

(p  z= 


COS.2a^(l fi2  sin.2  a^  (792) 

[(l+c2  cos.2")  (JT(we'V^i)-iz(?ie>o))-e^cos.'«(P' Vi-F'>o)], 
which  is  the  required  equation  of  the  curve. 

The  length  of  the  curve  becomes,  by  substitution  in  (781), 
8=  A  a/  (1— e2  sin.2  «)  {E'H^^—E'^^).         (793) 

The  integral  (790)  is  called  the  elliptic  integral  of 
the  third  order ^  and  admits  of  theorems  similar  to  those 
of  the  first  and  second  orders. 

2.  To  find  the  shortest  line  upon  the  prolate  ellipsoid. 


^    130.]  RECTIFICATION    OF    CURVES.  165 

Elliptic  integral  of  the  third  order  Quadrature. 

Ans.  Let  the  axes  of  the  ellipsoid  be  represented  by  the 
same  letters,  as  in  the  preceding  example  ;  and  let  the  equa- 
tions of  the  ellipsoid  be 

uzziB  COS.  &y  z:=A  sin.  &,  •\ 

Cz^B  COS.  a,      sin.  ^i=:sin.  «  sin.  V,  >     C^^^) 

e'=e  sin.  "^  nz=. — sin. 2  cc^  J 

the  equation  of  the  required  curve  is  ('95) 

e^  cos  « 
y=cos.«\/(l-c2)[^(ne>i)-72(nc>o)]  +  -^-j— ^(Fv^i-i^V^o), 

and  the  length  of  the  curve  is 

5  =  A  {E'y^i  —  E'%).  (796) 

3.  Prove  that  if  (pg,  (p  and  9'  satisfy  the  lovi^er  equation 
(599),  and  if 

N=A^[n{n+l){n  +  e'')]  (797) 

the   elliptic  integrals  of  the  third  order  will  satisfy  the  equa- 
tion 


n 


{n  e  (f')  -{-  n  (^n  e  (p)  —  n  {n  e  (fo) 


^  ♦      r-n  /  Ns\n.  cp'  sm.  <p  sin.  %  \ 

=z  ir^tan.^  ^J  I  -r—. — r—  I.     ((98) 

N  \   i  -\-n  (i  —  cos.  (f'  cos.  tp  cos.  Vo)    / 


166  INTEGRAL    CALCULUS.  [b.  V.   CH.  VII. 

Area  of  a  plane  surface. 


CHAPTER   VTI. 

QUADRATURE    OF    SURFACES. 

131.  Problem.     To  find  the  quadrature  of  a  surface. 

Solution.  Let  M  L  M'  L'  (fig.  7)  be  the  portion  of  the  sur- 
face, whose  area  is  to  be  found,  and  which  may  be  either  plane 
or  curved.  Let  the  conditions  of  the  bounding  line  be  ex- 
pressed by  an  equation  between  two  variables,  I  and  m.  Sup- 
pose two  lines,  L  L\  ll\  drawn  infinitely  near  each  other,  and 
in  such  a  way  that  I  is  constant  throughout  the  extent  of  these 
lines ;  and  let  the  lines  MM',  m  m\  be  so  drawn  that  m  is 
constant  throughout  their  extent.  If  then  <^  is  taken  to  denote 
the  required  surface,  we  have 

the  area  Z- Z.' n'  L  ,>^r^r^s 

^'"^ IT ^UT  ('^') 

and 

1)2      o  __  ^ n  L  _  the  area  ah  c  d  .^^^. 

'•"     ~      dl     ^      dl.dm         '  ^        ' 

But,  if  a  z=i  the  angle  h  a  c,  ^ 

5' =  an  arc  of  Z  Z',  >     (801) 

s"  =  an  arc  of  3Im ;  j 

we  have         the  arc  a  b  z=  d s',  the  arc  a  c  =:  ds",  (802) 

the  area  ab  c  dz=i  sin.  a  d  s'  .  d  s' 

and,  since  m  is  the  only  variable  in  s',  and  /  the  only  variable 


m  s' 


D],^o  —  sin.  aD^s' .  D^  s",  (803) 


«5>    134.]  QUADRATURE    OF    SURFACES.  167 

Area  of  a  curved  surface. 

and  the  accents  may  be  omitted  in  (802)  without  any  ambi- 
guity.    Hence 

c  =ifj,,  sin.  aD^.s,  Di.s;  (804) 

in  which  D^s  and  DiS  may  be  taken  directly  from  the  gen- 
eral expression  for  D  s,  and  a  is  the  inclination  of  two  lines 
drawn  through  a  point,  in  such  a  way,  that  for  the  one  /  is 
constant,  and  for  the  other  m  is  constant. 

132.  Corollary. ^  When  the  surface  is  plane,  (570)  of  vol.  1 
gives  for  rectangular  coordinates, 

l>,s=l,         />,  s=l,  (805) 

and  it  is  obvious  that  a  is  a  ri^ht  angle  ;  whence 

«=/,/,.!=/,.  a:  =/..y,  (806) 

or  supplying  the  place  of  arbitrary  constants  by  the  form  of 
definite  integrals, 

"  =fi  fi: '  =/::  ^^^-y^)  =f'i:  (^.-^»)'  (^«') 

in  which  the  values  of  a-^  Xy  7/0^1,  ^^^  determined  by  the 
bounding  curve. 

133.  Corollary.    When  the  surface  is  plane,  (574)  of  vol.  1 

gives 

DcfS=r,         DrS=\,  (808) 

and  «  is  a  right  angle  ;  whence 

o  =.f^f^  ^r^f^,r^P=^  if^p  .  r^•  (809) 

or 

134.  Corollary.    When   the  surface  is  curved,  let  Y  denote 
the  inclination  of  the  tangent  plane  to  the  plane   of  x  y,   and, 


168  INTEGRAL    CALCULUS.  [b.  V.   CH.  VII. 

Area  of  a  curved  surface. 

since  the  projection  of  a  surface  is  equal  to  the  product  of  the 

surface  by  the  cosine  of  its  inclination  to  its  projection,  (806) 

gives 

o=fJ^.sec.Y.  (811) 

Hence,  by  (600)  of  vol.  1,  where 

V—0  (812) 

is  the  equation  of  the  surface,         * 

=  fjy  ■  V  (D.  z^  +  ^,=='  +  !)•  (813) 

135.  Corollary.  When  the  surface  is  developable,  it  may 
be  supposed  to  be  developed  into  a  plane,  and  its  area  found  as 
that  of  a  plane  surface;  or  it  must  give  the  same  result  to 
refer  the  surface  to  axes,  drawn  upon  it  in  such  a  way,  that 
they  would  be  straight  lines  when  the  surface  was  developed, 
and  the  rectangular  coordinates  would  then  be  the  length  of 
the  shortest  lines,  which  would  be  drawn  upon  the  surface  to 
two  of  these  axes,  which  would  be  perpendicular  to  each 
other. 

136.  Corollary.  When  the  surface  is  one  of  revolution,  the 
notation  of  §  129  gives,  by  §  134, 

a^f,^f^.u^{D^z'^+\);  (814) 

and  if  5  denotes  the  arc  of  the  generating  curve, 

""  =hfu  'U  DuS  =fcpf,  ,uD,.s  =fcff,  .  u.     (815) 

137.  Corollary.  When  the  surface  of  revolution  is  included 
between  four  curves,  of  which  two  are  the  intersections  with 
the  surface  of  two  planes  which   are  perpendicular  to  the  axis 


«§>  140.]        QUADRATURE  OF  SURFACES.  169 


Quadrature  of  a  surface  of  revolution. 


of  revolution,  and  the  other  two  are  the  intersections  with  the 
surface  of  the  planes,  which  may  be  called  meridian  planes, 
because  they  include  the  axis  of  revolution,  and  which  are  in- 
clined to  each  other  by  an  angle  Vg  j  (^1^)  gives 


138.  Corollary.  If  another  surface  of  revolution  were  gen- 
erated by  the  revolution  of  the  arc  in  the  preceding  section, 
about  an  axis  at  the  distance  h  from  the  former  axis,  and 
farther  from  the  arc,  so  that  for  this  new  axis  we  have 

u'  =  u  +  h,  (817) 

(816)  gives  the  value  of  the  corresponding  surface 

o'  =  ^^fl^^  {u  D^^  s  +  b  D,,  s) 

—  a^bcp^  (Si—So).  (818) 

139.  Corollary.  Had  the  second  axis  been  upon  the  oppo- 
site side  of  the  arc,  we  should  have  had 

u"  z=  b  —  u  (819) 

o"  =  b^,{s,-s,)-o.  (820) 

140.  Corollary.  A  curve  AB  A'  B'  (fig.  8)  is  said 
to  have  a  centre  0  when  there  is  such  a  point  that  any 
chord,  such  as  A  A\  B  B',  &c.  which  passes  through 
it,  is  bisected  by  it ;  and  such  a  chord  is  called  a  diam- 

15 


170  INTEGRAL    CALCULUS.  [b.  V.  CH.  VII. 

Surface  of  a  ring. 


eter.  The  surface  generated  by  the  revolution  of  such  a 
curve  about  an  axis  C  C'  which  does  not  intersect  the 
curve,  is  called  mi  angular  surface^  or,  simply,  a  ring. 

The  notation 

S  =z  the   perimeter   of  the    generating   curve 
ABD  AD'  A, 

o  =:  the  surface  which  would  be  generated  by 

the  revolution  o(  D  B  A  D'  about  the  di-   ^    (821) 
ameter  D  D'  parallel  to -CC", 

h  z=.  the  distance  of  the  axis  C  C'  from  the  cen- 
tre, 

gives  by  (818  and  820)  for  the  whole  surface  of  the  ring, 

=:2b^  S.  (822) 

141.  Problem.    To   transform  the  differential  coefficient  of 
a  surface  from  one  system  of  variables  to  another. 

Solution.    Let  /  and  m  be  the  given  variables,  and  let  the 
second  member  of  (803)  be  denoted  by  //,  that  is, 

Dl^.o^zH  (823) 

If,  then,  only  one  of  the  variables  m  is  to  be  changed,  and 
t  is  to  be  introduced  instead  of  it  by  means  of  the  equation 

M  =  m,  (824) 

in  which  M  is  a  given  function  of  I  and  t ;  we  have 
D].,.o  —  D,D,.o^D^D,.o.D,.m 

—  Dl^.G  .  D,3I=z  H  D,.M.         (825) 


«J  144.]  Q,UADRATURE    OF    SURFACES.  171 


Transformation  of  differential  coefficient. 


If  the  Other  variable  /  is  also  to  be  changed,  and  u  to  be  in- 
troduced instead  of  it,  by  means  of  the  equation 

/  =  L,  (826) 

in  which  Z«  is  a  given  function  of  t  and  u  ;  we  have 

D]  ,o^HD,.M.D^L,  (827) 

142.  Corollary.  If  J/,  in  equation  (824),  instead  of  be- 
ing a  given  function  of  t  and  /,  were  a  given  function  of  t  and 
u,  u  might  be  eliminated  by  means  of  (826).  It  is  more  con- 
venient, however,  to  eliminate  its  differential  coefficient  only 
from  Di .  31,  after  having  determined  this  differential  coef- 
ficient by  means  of  (826).  Thus  the  differential  of  (826) 
relative  to  t  is,  by  regarding  w  as  a  function  of  t, 

0  =  D,L  +  D,L  .  D,u,  (828) 

whence  ^^  ,  ^  _  g^  ,  (829) 

and  (824)  gives 

=  —j^^ .     (830) 

But  771  is  obviously  to  be   substituted  for  31  in  (827),  whence 
we  have  by  (827  and  830), 

Dl^  azzzH  (D,3I.D^,L—D,,3I.D,L).     (831) 

143.  Corollary,  The  two  preceding  articles  may  be 
applied  to  the  transformation  of  any  second  differential 
coefficient  of  two  successive  variables. 

144.     Examples. 

1.  To  find  the  area  of  the  segment  of  an  ellipse  included 
between  two  parallel  lines. 


172  INTEGRAL    CALCULUS.  [b.  V.  CH.  VII. 

Transformation  of  differential  coefficient. 


Solutio7i.  Let  the  ellipse  be  referred  to  conjugate  axes,  as 
in  (74)  of  vol.  1,  in  which  the  axis  of  ?/  is  drawn  parallel  to 
the  given  lines  ;  and  (804  and  807)  give,  since  in  this  case 

y„  =  -y,  (832) 

is  the  ordinate  y  of  the  ellipse,  if  «  is  the  angle  of  the  axes 

0  =  2  f""^  .  7/ sin.  «.  (833) 

If,  now,  we  take  5  so  that 

X  ^^  A  cos.  5,  (834) 

we  have  y  z=.  B  sin.  ^,  (835) 

D^,x  =  —  A  sin.  ^,  (836) 

a=2sin.  «  r  ^  A  B  sm.^ &z=zA  J5  sin.  «  /"  0  (i  —  cos.  2  &) 

J  &i  */   ^1 

=^jB  sin.  a  [6^—6^—^{s\n.  2  ^q— sin.  2  ^  J]  (837) 

=■  J  sin.  «  (corresponding  areaof  asegmentof  a  circle  whose 
radius  is  yl). 

2.  To  find  the  area  of  a  sector  of  an  ellipse,  when  the  ver- 
tex of  the  sector  is  at  the  centre  of  the  ellipse. 

Solution.  In  this  case  (834  and  835)  give,  when  A  and  B 
are  the  semiaxes, 

r  COS.  (p  =z  A  cos.  Q, 

B 

tan.  (p  zzz  —  tan.  &, 
A 

t^Dq.  pzzzA  B  ; 

whence,  by  (810),  putting  zero  for  r^  , 

c  =  iAB(\-i,).  (841) 


r  sin.  (f  z=  B  sin.  6, 

(838) 

^               B  C0S.2  (p 
T)      (p  — 

^           A  C0S.2  a  ' 

(839) 

(840) 

<5>    144.]  QUADRATURE    OF    SURFACES.  173 

Area  of  elliptic  segment  and  sector. 


Corollary.    The  whole  area  of  the  ellipse  \s,  n  A  B.     (842) 

3.  To  find  the  area  of  a  sector  of  an  ellipse,  when  the  ver- 
tex of  the  sector  is  at  a  focus. 

Solution.  If  the  origin  of  coordinates  is  at  the  focus,  (834 
and  835)  give 

r  cos.  (f=iX7^A  cos.  ^ — A  e=A  (cos.  & — e)  (843) 

r  sin.  (fz:zi/^zzB  sin.  &  (844) 

B  sin.  &  ,r.4^. 

D6,^?.  (lz:i££!_!L-^^  (S46) 

A  (cos.  <3 — e)2  ^      ^        ' 

r^  D6cp  =  AB  .  (1—  e  cos.  &),  (847) 

whence,  by  (810), 

<^=zA  B  .  [6^—6^—e  (sin.  ^^— sin.  dj].  (848) 

4.  To  find  the  area  of  the  hyperbolic  segment  included  be- 
tween two  parallel  lines. 

A?is.  If  the  hyperbola  is  referred  to  conjugate  axes  as  in 
(90)  of  vol.  ],  in  which  the  axis  of  ?/  is  parallel  to  the  given 
lines,  if  y  is  the  angle  of  the  axes,  and  if  ^  is  taken  so  that 

xz=  A  Cos.  ^      y  =  B  Sin.  ^,  (849) 

the  area  is 

a  =  J  ^JBsin.y  (Sin.  2^j— Sin.  2  ^0+2  ^1—2  5 J.  (850) 

5.  To  find  the  area  of  the  hyperbolic  sector,  the  vertex  of 
which  is  at  the  centre  of  the  hyperbola. 

15* 


174  INTEGRAL    CALCULUS.  [b.  V.   CH.  VII. 

Area  of  hyperbolic  segment  and  sector. 

Ans.    With  the  notation  of  the  preceding  example,  the  area 

"^^B  ("-«„)■  (851) 

in  which  A  and  B  are  the  semi-axes, 

6.  To  find  the  area  of  the  Ijiyperbolic  sector,  the  vertex  of 
which  is  at  one  of  the'  foci. 

Ans.  With  the  notation  of  the  preceding  example,  the  area 
is 

oz=AB  [(^— ^o)  —  ^  (Si»-  ^— Sin.  d  J].      (852) 

7.  To  find  the  hyperbolic  segment  included  between  an 
asymptote,  the  curve,  and  two  straight  lines  drawn  parallel  to 
the  other  asymptote. 

Solution.  It  is  convenient,  in  this  case,  to  take  the  two 
asymptotes  for  the  oblique  axes,  for  which  «  and  ^  in  (86)  of 
vol.  I.  must  have  the  values 

tan.  a  z=  -   ,  tan.  ^  =  —  -  ;  (853) 

whence  (86)  gives  for  the  equation  of  the  hyperbola,  referred 
to  its  asymptotes, 

xy:=:zl{A^+B^).  (854) 

The  area  of  the  required  segment  is,  then,  by  (807,  853  and 

854),  if  the  axis  of  y  is  the  asymptote  parallel  to  the  given 

lines, 

.     ^      px.  2AB     px. 


^  «/    3  0   ^ 


JAi3  1og.  ^.  (855) 


<5>    144.]  Q,UARDATURE    OF    SURFACES.  175 


Area  of  parabolic  and  cycloidal  segments. 


8.  To  find  the  area  of  the  parabolic  segment  included  be- 
tween two  parallel  lines. 

Ans.  If  the  parabola  is  referred  to  oblique  axes  as  in  (100) 
of  vol.  I,  of  which  the  axis  of  3/  is  parallel  to  the  given  lines, 
and  if  «  is  the  angle  of  the  two  axes,  the  area  is 

""^l  (i/i  2:1— yo  2:0).  (856) 

9.  To  find  the  area  of  the  parabolic  sector,  of  which  the 
vertex  is  at  the  focus. 

Ans.  If  P  is  the  distance  from  the  vertex  to  the  focus,  if 
the  origin  is  at  the  focus  and  the  angle  y  counted  from  the 
vertex,  the  area  is 

0  —  2P  (tan.  J  ^1— tan.  J  </'o).  (857) 

10.  To  find  the  area  of  the  segment  included  between  the 
curve,  the  axis  of  x,  and  two  lines  drawn  parallel  to  the  axis 
of  y,  of  the  curve  known  as  the  parabola  of  the  order  a,  which 
has  for  its  equation 

2/  =  Ax\  (858) 

Ans.    o=^J^rZhlo_,       (859) 

11.  To  find  the  are  of  the  segment  of  a  cycloid  given  by 
equations  (130  and  131)  of  vol,  1,  included  between  the  curve, 
the  axis  of  x,  and  two  lines  drawn  parallel  to  the  axis  of  y. 

Ans.  ^=722[|.(^,-5o)-2(sin.^-sin.^o)+l-(sin.2^i-sin.2  6o)].  (860) 

Corollary.  The  whole  area  included  between  a  branch  of 
the  cycloid  and  the  axis  of  z,  is 

C  =   ^7VR^^  (861) 

=  three  times  the  area  of  the  generating  circle. 


176  INTEGRAL    CALCULUS.  [b.  V.   CH.  VII. 

Area  of  sectors  of  spirals. 

12.  To  find  the  area  of  the  segment  of  a  cycloid,  which  is 
included  between  the  curve  and  a  line  drawn  parallel  to  the 
axis  of  y. 

Ans.    o—R^[(^n—&)  (1+2  cos.  (3)_2sin.  a— Jsin.  2^]. 

13.  To  find  the  area  of  a  sector  of  the  spiral  of  equation 
(133)  of  vol.  1,  when  the  vertex  of  the  sector  is  at  the  origin. 

14.  To  find  the  area  of  a  sector  of  the  hyperbolic  spiral,  the 
equation  of  which  is  (135)  of  vol.  I,  when  the  vertex  of  the 
sector  is  at  the  origin. 

Ans.    o  —  2n'^R^  /_i___iy         (864) 

15.  To  find  the  area  of  a  sector  of  the  logarithmic  spiral,  of 

which  the  equation  is 

r  =  a  e     ,  (865) 

when  the  vertex  of  the  sector  is  at  the  origin. 

Ans.    o^^a'^ie    ^'  —  e      °).      (866) 

16.  Given   the  area  <^  of  a  surface  included   between   any 

lines  whatever,  the  combination  of  which  considered   as  one 

line  which   in  general  is  discontinuous,  is  represented  by  the 

equation 

F.{T,y)=0,  (867) 

to  find  the  area  a  of  the  surface  bounded  by  the  line  or  system 
of  lines 

Ans.    a'— aha.  (869) 


<§>    144.]  QUADRATURE    OF    SURFACES.  177 

Area  of  a  zone  of  an  ellipsoid. 

Corollary.  If  a  and  h  are  equal,  the  surfaces  are  similar, 
and  (869)  gives 

o'=a^a-  (870) 

that  is,  the  areas  of  shnilar  surfaces  areproportional  to 
the  squares  of  their  dimensions. 

17.  To  find  the  area  of  the  zone  of  an  oblate  ellipsoid  of 
revolution  which  is  included  between  two  planes  drawn  per- 
pendicular to  the  axis  of  revolution. 

Solution.  Let  the  notation  be  that  of  Example  1,  of  §  130, 
and  (816)  gives,  for  the  area, 

7  ^^^^^ 

Let  the  angle  w  be  so  taken  that 

B  Sin.  ^==  Ac  COS.  6 ;  (872) 

and  we  shall  have 

—  Ae  sin.  6  Doj.^^B  Cos.  oj;  (873) 


whence 

2  -ft  jB2 


/: 


'°  .  COS.2 


e    .,        1 


=  - /       °    .   (1+CoS.  2  a,) 


=  ^^  [(%--i)  +4  (Sin.  2c.^-Sin.2  .J].      (874) 

18.  To  find  the  area  of  the  zone  of  a  prolate  ellipsoid  of 
revolution  which  is  included  between  two  planes  drawn  per- 
pendicular to  the  axis  of  revolution. 


178  INTEGRAL    CALCULUS.  [b.   V.  CH.  VII. 

Area  of  a  zone  of  a  hyperboloid. 

A/is.    With  the  notation  of  Example  I,  of  §  130,  and  put- 
ting 

COS.  oj  ^=  e  cos.  &,  (875) 

the  area  is 

"  =  "^^  [(-  --'o)  +  i  (sin-  V  -s'»-  2  %)]•  (876) 


19.  To  find  the  area  of  the  zone  of  the  hyperboloid  of  revo- 
lution formed  by  the  revolution  of  an  arc  of  an  hyperbola 
about  the  transverse  axis, 

Ans.  If  the  equations  of  the  generating  hyperbola  are 

xz=z  A  Cos.  ^        !/  =  B  Sin.  &,  (877) 

and  if  w  is  taken  so  that 

e  Cos.  &  z=z  sec.  «»,  (878) 

the  area  is 

_nABrsm.'^^       sin.o'o     ,  ,_  tang.  (45°+^  co  )-| 

^-^-  L^^r^^'o+^'t^^ii:(45^+i^)J-^ 

20.  To  find  the  area  of  the  zone  of  the  paraboloid  of  revo- 
lution, included  between  two  planes,  which  are  perpendicular 
to  the  axis  of  revolution. 

Ans.    If  P  is  the  distance  from  the  vertex  to  the  focus,  and 

if  &  is  so  taken  that 

y  =  2P  tan.  5,  (880) 

the  area  is 

a  =  f  T  P2  (sec.s^i— sec.^dj.  (881) 

21.  To  find  the  area  of  the  zone  generated  by  the  revolution 
of  an  arc  of  a  parabola  about  the  axis  of  ?/  of  the  preceding 
example. 


«§>   145.]  QUADRATURE    OF    SURFACES.  179 

Area  of  a  zone  generated  by  the  arc  of  a  cycloid. 

Ans.    If  ^  is  taken  so  that 

X  -{-  P  =  P  sec.  6,  (882) 

and  if  o'  is  the  value  of  o  in  (879), 
the  area  is  P2  e 


A  B 


(883) 


22.  To  find  the  area  of  the  zone  generated  by  the  revolu- 
tion of  an  arc  of  a  cycloid  about  the  axis  of  x  in  (130)  of 
vol.  1.     The  arc  is  supposed  to  commence  with  ^. 

A?is.  With  the  notation  of  equations  (130  and  [31)  of 
vol.  1,  the  area  is 

G=l6  7tR2  {2_2  COS.  i  5  —  ^  sin.2  J  6  .  cos.  J  5).    (884) 

23.  To  find  the  area  of  the  zone  generated  by  the  revolu- 
tion of  an  arc  of  a  cycloid  about  the  axis  of  y  in  (131)  of 
vol.  1.     The  arc  is  supposed  to  commence  with  6. 

Ans.  With  the  notation  of  the  preceding  example,  the  area 
is 

a  =  16  nR2  (sin.  J  ^— J  &  COS.  J  ^  —  ^  sin:3  J  6).      (885) 

145.  Problem.  To  find  the  area  of  the  zone  gen- 
erated by  the  revolution  of  a  given  arc  of  a  plajie  curve 
about  an  axis  in  the  same  plane  with  the  arc^  when  the 
areas  of  the  two  zones  are  known  which  are  generated 
by  the  revolution  of  the  arc  about  two  axes  in  the  plane, 
which  are  perpendicular  to  each  other. 

Solution.  Let  the  two  perpendicular  axes  be  those  of  x 
and  y,  and  let  the  given  areas  be,  by  (816), 


180  INTEGRAL  CALCULUS.      [b.  V.  CH.  VIL 

Greatest  or  least  surface. 


o'=r  2  TT 


o"—2 


r^'.x.  (887) 


Let  the  new  axis  be  inclined  to  the  axis  of  a;  by  an  angle  «,  and 
pass  at  a  distance  a  from  the  origin,  and  the  required  area  is 

0=1:^2^  f    ^ '  {y  eos.  «  —  X  sin.  «  —  a) 
—  _j_  2  TT  [a'  cos.  «  —  o"  sin.  a  — a  {s^—s^)],  (888) 

in  which  that  sign  is  to  be  adopted  which  renders  the  second 
member  positive. 

146.  Problem.  To  draw  the  curve  line  subject  to 
given  cojiditionSj  which  includes  a  maximum  or  mini- 
mum surface. 

Solution.  This  problem,  like  that  of  §  116,  involves  the 
maximum  or  minimum  of  a  definite  integral,  and  is  therefore 
solved  in  a  similar  way,  by  the  method  of  variations.  There 
is,  in  this  case,  however,  a  double  integral,  and  the  first  inte- 
gral refers  evidently  not  to  disconnected  points,  but  to  the 
bounding  lines  of  the  surface,  so  that  the  determination  of 
these  lines  may  involve  the  method  of  variations,  even  when 
the  general  form  of  the  surface  is  given.  The  determination 
of  the  form  of  the  surface  will  admit  of  more  lucid  dis- 
cussion in  a  chapter  upon  the  curvature  of  surfaces,  and 
the  present  chapter  will  be  confined  to  the  consideration  of  the 
bounding  line. 

The  equation  of  the  surface  being  given,  the  form  of  its 
second  differential  coefficient  is  known,  and  is  independent  of 


<§>  148.]         QUADRATURE  OF  SURFACES.  181 


Greatest  or  least  surface. 


the  limiting  lines,  so  that  an  integration  can  be  directly  per- 
formed, and  the  required  integral  be  reduced  to  the  form  (756), 
and  the  process  of  finding  the  maximum  or  minimum  becomes 
identical  with  that  of  §  1 IG. 

147.  Corollary.  A  kind  of  equntion  of  condition  is  often 
connected  with  this  problem,  wholly  different  from  those  refer- 
red to  in  §  116.  Each  of  the  equations  (754)  is  an  equation 
which  is  satisfied  by  the  coordinates  of  each  point  of  the  re- 
quired curve,  and  is  thus  equivalent  to  an  infinite  number  of 
equations.  But  an  equation,  of  the  class  here  alluded  to,  is  a 
single  equation,  involving  the  coordinates  of  every  point  of  the 
curve.  An  instance  of  such  an  equation  is  the  one  which 
expresses  that  the  bounding  curve  must  be  of  a  given  length, 
or  that  the  definite  integral  (741)  must  have  a  given  value. 

All  equations  of  this  kind  would  appear  to  depend,  neces- 
sarily, upon  definite  integrals,  and  they  may  be  introduced  into 
the  equation  of  maximum  or  minimum  for  the  purpose  of  elim- 
ination by  the  method  of  §  119.  It  must  be  observed,  how- 
ever, that  the  multipliers  ;.,  //,  &c  ,  of  these  equations  arc 
always  constant.  For  each  of  these  equations  does  not  deter- 
mine any  relation  between  $r,  $y,  &c.  which  is  applicable  to 
each  point  of  the  curve,  but  only  a  particular  relation  by  which 
one  of  the  variations,  as  ^x,  may  be  determined  for  one  of  the 
points  in  terms  of  the  values  of  the  variations  for  all  the  points. 
The  corresponding  multiplier  '',  therefore,  must  have  that  par- 
ticular value  which  shall  cause  this  single  value  of  ^x  to  disap- 
pear from  the  equation  ;  that  is,  ;.  must  be  constant. 

148.    Examples. 

1.  To  find  the  plane  curve  which,   having   a   given  length, 
encloses  the  maximum  area. 
16 


182  INTEGRAL    CALCULUS.  [b.  V.   CH.  VII. 

Greatest  or  least  surface. 


Solution.    The  function  to  be  a  maximum  is,  by  (806), 

'^  y,  (889) 


and  the  function  (566)  is  to  be  constant.  Hence  if  A  is  the 
constant  multiplier  introduced  for  the  purpose  of  elimination, 
the  equation  is,  by  the  reduction  of  §  121, 

1  -  ^  D.  (g^  )  =  0,  (890) 

or  by   the  notation  of  §  148  of  B.  II.,  and  by  (577  and  609 

of  vol.  1, 

0  =  1+^1),  .COS.  „  (891) 

0  =  DrX  -\- A  Dr  COS.  v 

—  sin.  r  Dys  —  A  sin.  ^  (892) 

Az:z  DrS  —  Q;  (893) 

that  is,  the  curvature  is  constant,  which  is  the  property  of  no 
Other  curve  than  the  circle ;  the  required  curve  is,  therefore,  a 
circle  ;  which  has,  already,  been  proved  in  the  Elements  of 
Geometry. 

2.  To  find  the  plane  curve  which,  being  drawn  from  one 
given  point  to  another  given  point,  and  having  a  given  length, 
encloses  the  maximum  area  between  the  curve  itself,  its  two 
extreme  radii  of  curvature  and  its  evolute. 

Solution.  By  adopting  the  notation  of  the  preceding  article, 
the  required  area  may  be  expressed  in  the  form 


1 


.  e~  ;  -  (894) 

0 

that  of  the  arc  will  be 


s 


=y  ;:■ .  c.  (895) 


§    148.]  QUADRATURE    OF    SURFACES.  183 

Greatest  or  least  surface. 

Equations  (576,  577  and  009)  of  vol.  1,  give 

Di  X  z=  sin.  r  Di  5  =  o  sin.  ,,  (S9G) 

Dy  7/  Z=Z  COS.  V  Di  S  = n  COS.  i .  (897) 

The  given  differences  of  the  coordinates  of  the  extreme  points 
of  the  curve  are,  then, 

X,  —  2,  =  fl'  .  Q  sin.  r,  (89S) 

ft/       '  0 

!/i—!/o——t   ' '  •  ^  COS.  ,.  (899) 

If,  therefore,  A,  B,  C  are  the  constant  multipliers  of  (895, 
898  and  899),  introduced  for  the  purpose  of  elimination,  the 
equation  of  the  maximum  or  minimum  is 

o  o  _|-  ^  -|_  J5  sin.  r  —  C  COS.  r  =  0.  (900) 

Let  H  and  «  be  taken  so  that 

B z=zH COS.  cc^         C=Hsm.ai  (901) 

and  (900)  becomes 

2q  +  A  +Hsm.  (»'  — a)  =  0;  (902) 

and  by  putting 

r'  =  v  — «,  (903) 

2Q  +  A  +  Hsin.r'  =  0;  (904) 

which  shows  that  (900)  may  be  reduced  to  the  form  (904), 
from  which  the  term  containing  cos.  ^  disappears,  by  merely 
changing  the  direction  of  the  axis  of  x.  It  does  not,  then, 
diminish  the  generality  of  the  solution  to  put 

0=0;  (905) 

by  which  (900)  becomes 

2  §  +  ^  +  J5  sin.  V  =  0.  (906) 


184  INTEGRAL    CALCULUS.  [b.   V.   CH.   VII. 

Greatest  or  least  surface. 

Tiie  curve  is  easily  expressed  in  rectangular  coordinates  by 
the  equations 

2  =  ^  A  COS.  r  -\-l  B  sin. 2  '  +  ^  r,  (937) 

ij  —  ^A  sin.  '■  —  ^  B  COS.  2  »•.  (908) 

Corolla nj.    When   the   extreme    points   are   not    fixed,  tlie 
equation  (900)  becomes 

'2  c  +  ^  =  0  ;  (909) 

that  is,  the  curve  is  a  circle. 

Corollary.     When  the  length  of  the  curve  is  not  given,  the 
equation  (906)  becomes 

"Zq  +  B  sin.  1=0;  (910) 

which  is,   evidently,    from    example  3  of  §  151  of  B.  II.,  a 
cycloid. 


<5>  149.]  CURVATURE    OF    SURFACES.  185 

Curvature  of  a  surface  in  any  direction. 


CHAPTER   VIII. 


THE  CURVATURE  OF  SURFACES. 

149.  Problem.  To  find  the  curvature  of  a  given 
surface  at  any  point  in  any  direction. 

Solution.  Let  the  tangent  plane  to  the  surface  at  any  one 
of  its  points  be  taken  for  the  plane  of  the  coordinates  x  and  y, 
so  that  the  normal  may  be  the  axis  of  z.  We  have,  then,  at 
this  point, 

D,z=0,  D,^z  =  0;  -(911) 

and  if  q^  and  c^  are  the  radii  of  curvature  at  the  point  of  the 
intersections  of  the  planes  ofocz  and  1/ z  with  the  surface, 
equation  (610)  of  vol.  1  gives 

-^^Dlz,        ^-=I>lz.  (912) 

The  radius  of  curvature  (?  of  a  section  made  in  any  intermedi- 
ate direction  by  a  normal  plane,  which  is  inclined  to  the  axis  of  x 
by  the  angle  «,  is  derived  from  the  equation 

—  =  Bl  z,  (913) 

if  u  denotes  the  distance  of  a  point  of  the  curve  of  intersec- 
tion from  the  axis  of  z.  But  the  coordinates  of  one  of  these 
points  are 

xz=iu  cos.  «,     y  =^u  sin.  «  ;  (914) 

16* 


186  INTEGRAL  CALCULUS.       [b.  V.  CH.  VIII. 

Directions  of  greatest  and  least  curvature. 

whence,  in  general, 

D^  z  =  COS.  »  D^z  -j-sin.  ^  DyZ,  (915) 

-=zDlz  r=  COS.- a  Biz  4-2  sin.  «  cos.  «  B'l_yZ  -\-  sin.^a  Dl  z 

C0S.2  a    ,   sin.~  «      ,    ^    .  ^o 

= 1 1- 2  sin.  "COS.  oc  Dl,y  z.  (916) 

150.  Corollary.    The  radius  of  curvature  (/,  in  a  direction 
perpendicular  to  that  of  ^,  is  given  by  the  equation 

1       sin."«  ,    cos.2«      ^    .  _. 

—=: 2  sin.  «  COS.  a  D'i   y  z.        (91  / ) 


151.  Corollary.    The  sum  of  (916  and  917)  is 

-+T  =  -+-;  (91S) 

that  is,  the  sums  of  the  reciprocals  of  the  two  radii  of 
curvature  of  any  two  perpendicular  sections  at  a  given 
point  of  a  surface  is  a  constant  quantity. 

152.  Corollary.  If  Q  were  the  maximum  radius  of  curva- 
ture at  the  point,  o'  would  obviously  be  the  minimum  radius  of 
curvature  ;  whence 

The  directions  of  greatest  and  least  curvature  of  a 
surface  at  any  point  are  perpendicular  to  each  other. 

153.  Corollary.    The  difference  between  (916  and  917)  is 
—  —  i=:cos.2«  f -  — -^  — 2sin.2«Z>|..  ^,        (919) 

Q'  Q  \Qy         Qz    / 

and  in  the  hypothesis  of  the  preceding  corollary,  the  first  mem- 


<§>    155.]  CURVATURE    OF    SURFACES.  1S7 

Motion  of  point  of  contact  in  direction  of  greatest  or  least  curvature. 

ber  of  (919)  is  a  maximum.  The  differential  coefficient  of 
the  second  member,  taken  with  reference  to  «,  must  be  equal 
to  zero,  that  is,  , 

0  =  sin.  2  '^(-  — ,7  )  +  2  cos.  2  a  D\.yZ.        (920) 

The  sum  of  (919)  multiplied  by  cos.  2  «,  and  of  (920)  Uy 
sin.  2  a,  is 

cos.2«(^--^=:---.  (921) 

\    ''  'I  /  "  Or 

Hence,  from  (918), 

cos.2  «       sin.'2  « 


from  which  the  curvature  of  the  surface  can  be  found 
in  a  direction  inclined  by  the  angle  «  to  the  direction  of 
maximum  curvature. 

154.   Corollary.    One  half  of  the  difference  between  (919) 
multiplied  by  sin.  2  «,  and  (920)  by  cos.  2  «,  is 


D-j,.y  z  z=  —  1  sin.  2  « 


Q-D-    '-' 


155.   Corollary.    For  the  direction  of  the  maximum  or  min- 
imum, «  is  zero  or  a  right  ajigle,  and,  therefore,  for  either  of 

these  directions, 

D%.yz  =  0-  (924) 

that  is,  with  a  small  motion  of  the  point  of  contact  in 
the  direction  of  the  greatest  or  least  curvature,  the  tan- 
gent plane  rotates  about  a  line  perpendicular  to  the 
direction  of  the  motion  of  the  point. 


188  INTEGRAL    CALCULUS.  [b.  V.  CH.  VIII. 

Direction  of  no  curvature. 

156.  Corollary.  When  «  is  half  a  right  angle,  (921  and 
922)  give 

Qy  =  Qx,  (925) 

l=l=Wi+M.  (926) 

157.  Corollary.  When  the  values  of  Q  and  (?'  have  opposite 
signs,  neither  of  the  corresponding  curvatures  is  strictly  a  min- 
imum, but  the  two  curvatures  are  the  greatest  curvatures  in 
opposite  directions.  There  are,  in  this  case,  two  intermediate 
directions  of  no  curvature,  corresponding  by  (922)  to  the 
values  of  «, 

tang.  «r=±\/(— ^yV  (927) 

The  sections  of  the  surface,  made  in  these  directionSj 
have  a  contact  of  the  second  order  with  the  tangent 
plane,  and  correspond,  in  general,  to  points  of  contrary 
flexure. 

158.  Corollary.  In  the  case  of  a  point  of  contact  for  which 
the  greatest  and  least  curvatures  are  in  opposite  directions  and 
equal,  we  have 

Q^-Q';  (928) 

whence,  by  (918), 

Cz  =  —  (?,;  (929) 

that  is,  the  curvatures  iu  any  two  directions,  which  are 
perpendicular  to  each  other,  are  equal  and  opposite. 

We  have  also  in  this  case,  by  (927), 

«  —  zb  45°  (930) 

for  the  angles,  which  the  directions  of  no  curvature  make  with 
the  direction  of  greatest  curvature. 


<§>    160.]  CURVATURE    OF    SURFACES.  1S9 


Curvature  of  a  section  which  is  not  normal  to  the  surface. 

159.  Corollary.  If  the  curvature  were  required  of  a  sec- 
tion, the  plane  of  which  did  not  include  the  normal,  it  might 
be  found  by  referring  the  surface  to  an  oblique  system  of  co- 
ordinates, of  which  the  tangent  plane  was  the  plane  of  xy, 
the  cutting  plane  that  of  xz\  the  axis  of  x  being  the  intersec- 
tion of  these  two  planes,  and  the  axes  of  y  and  z'  being  per- 
pendicular to  that  of  X.  This  system  might  be  obtained  from 
the  rectangular  one,  which  has  the  same  axes  of  2:  and  y,  but 
in  whicli  the  axis  of  z  is  the  normal,  by  putting 

&  =  the  inclination  of  the  axis  of  z  to  that  of  z' ,  (931 ) 

=  the  complement  of  the  inclination  of  the  given  plane  to 
the  tangent  plane, 

which  gives  2;  =r  2'  cos.  ^,  (9i.5'2) 

D:  z  =  Dlz'  cos.  <5  ;  (9:3:i) 

or,  by   putting  Q^  =z  the  radius  of  curvature  of  the  inclined 
section 

-=i  cos.  5,  (934) 

(1^  =  0  COS.  &.  (935) 

IGO.  Corollary.  If  the  axes,  in  the  preceding  corollary, 
were  rectangular,  that  of  y  being  perpendicular  to  the  given 
plane,  and  those  of  x  and  z  situated  in  any  way  whatever  in 
that  plane,  equation  (610)  of  vol.  1  gives 

L  —  - — -'  - — ^^  .  .  (936) 


If  we  put 


the  anufle  of  C'^  and  z 


y  =z  the  angle  of  o  and  z,  ^     v     ' ) 

and  observe  that  the  plane  of  i'  and  'jj  is  perpendicular  to  that 


190  INTEGRAL    CALCULUS.  [b.  V.   CH.  VIH. 

Curvature  of  any  point  of  the  surface. 

of  ;:;  and  z,  so  that  if  a  sphere  were  described  with  the  point 

of  contact  for  the  centre,  the  arcs  &,  t,  y  would  form  upon  the 

surface  a  right  triangle,  of  which  y  was  the  hypothenuse,  we 

have 

cos.  y  =:  cot.  r  COS.  &.  (938) 

But  the  comparison  of  (81 1  and  813)  gives 

sec.y  =  ^[\+(D,zy-  +  {D,zf],  .  (939) 

and  we  have,  obviously, 

sec.  Tz=V[l+(/>x2)2];  (940) 


whence 

1        1       sec.  r         Dl .  z 


Q        Q^      sec.  Y      l+(Z>x  zf 

Dl.z  1 


.   cos.  y 


1  +  {D.  zf  •  V  [1  +  {Dy  ^f+  {D.  ^f] 


■   (941) 


161.  Corollary.  The  curvature  of  a  section  of  the  surface 
made  by  a  plane  which  includes  the  axis  of  z,  and  is  inclined 
to  the  plane  of  2;  x  by  the  angle  £,  may  be  found  by  the  formula 

in  which     w  =  the  distance  of  any  point  of  the  section  from 

the  axis  of  z, 
whence 

a;=M  cos.  c,       y  ^=-u  sin.  s ;  (943) 

i>„a;=:cos. «,    X>„yz=sin.  «;  (944) 

D^ 2=cos.  t  .D^z-\-  sin.  «  Dy  z,  (945) 

Dl z=cos.2  a .  Dlz+'^  sin. « cos.  £  DJ .,2+sin.2  a .  I>2 ^  j  (946) 

1_   _  {Dlz  +  2  tan,  e .  Dl,yZ-\-i^n.^  s .  Dl  z)  cos,  y 

Q    "~l+Z>^22_|.2tan.eZ>,2;i>y2;+(l+i>,%2)tan.2e' ^       ^ 


<§>   163.]  CURVATURE    OF    SURFACES.  191 

Curvature  of  any  part  of  the  surface. 

and  since  the  coordinates  x,  y,  z  do  not  themselves  occur  in 
this  value  of  the  reciprocal  of  the  radius  of  curvature,  but 
only  their  differentials,  (947)  is  applicable  to  any  point  of  the 
surface,  and  to  any  direction  of  the  curvature,  it  being  ob- 
served that  s  is  the  angle,  which  the  plane,  dravi^n  through  the 
axis  of  z  and  parallel  to  this  direction,  makes  with  the  plane 
of  xy. 

162.  Corollary.  When  the  plane  which  is  parallel  to  the 
required  direction  of  curvature  is  also  parallel  to  the  radius  of 
curvature,  (601,  598  and  599)  of  vol.  1  give 

cos.    fi  Dy  Z  /rv40x 

tan.  £  = ■  =  — ^  ;  (948) 

COS.  «       IJ^  z 

whence   the  product  of  the  denominator  of  (947),  by  Dx  z^, 

becomes 

D^z^  ^  Dxz^  +  2D.  %2  DyZ^  +  DyZ^  +  D^z^ 

=  {D,  z^  +  Dyz')    (1  +Dzz2  +  Dy  z2) 

=  {D^z^  -{-  Dyz2)sec.2Y;  (949) 

and  (947)  becomes 

1       Drz^  Dlz+2  D,  z  DyzDl^y^z-^Dyz'Dlz 


Dx  z^+Dy  z' 


cos.3y.(950) 


163.  Corollary.  When  the  direction  of  curvature  is  per- 
pendicular to  that  of  the  preceding  article,  the  plane  which  is 
parallel  to  it  is  also  perpendicular  to  that  of  the  preceding 
article ;  whence,  in  this  case, 

Dxz 

tan.    i'  -Z  cot.   «   = =::^ —     , 

U  z 
and  (917)  becomes 


19-2  INTEGRAL    CALCULUS.  [b.  V.  CH.  Till. 


Sum  of  any  two  perpendicular  radii  of  curvature.    • 


1C4.    Corollary.    The  sum  of  (950  and  951)  is 
11^^  [\+DyZ')Dlz-^D.zD^zDl,,^z^{\+D^z^)Dlz 

'       ^      '  ""'^'  (952) 

which  is,  by  (918),  the  sum  of  the  reciprocals  of  the 
greatest  and  least  radii  of  curvature  at  the  point  x^  y,  z  ; 
or  it  is  the  sum  of  any  two  perpendicular  radii  of  cur- 
vature. 

165.  Problem.  To  find  the  greatest  or  least  surface 
which  can  he  drawn  under  given  conditions. 

Solution.  This  form  of  statement  embraces  that  portion  of 
the  problem  of  §  146  which  was  reserved  for  this  chapter. 
Since  a  single  equation  between  the  coordinates  of  each  point 
is  sufficient  to  determine  the  surface,  no  such  equation  can  be 
given  ;  but  there  may  be  particular  conditions  invohing  defi- 
nite integrals,  like  those  referred  to  in  §  J 46. 

166.  Corollary.  When  there  is  no  condition  what- 
ever, the  required  surface  is  absolutely  the  least  surface 
of  all  lohich  have  the  same  boundary. 

In  this  case,  the  integral  to  be  a  minimum  is  (Sli  or  813), 
the  variation  of  vrliich  gives 

ff^.  COS..,  {D,zDJz+DyzD^dz)=i^.         (953) 
But,  by  integration, 
fz  fy .  COS.  y  Dxz  D-^sz  =fyfz .  cos.  y  Dj,Z  D^Sz 

=/y  .  Dz  z  cos.  y  8  z—frfy  .  D:c  (cos.  Y  Dj^z)  s  z,    (954) 

(955) 
ffj .  cos.y  Dy  zDy  ^  z—fj.  .By  z  COS./'  (J  z-fxfy .  By  (cos.  yBy  zy  z ; 


<§>    167.]  CURVATURE    OF    SURFACES.  193 

Least  surface. 

whence,  by  regarding  only  the  terms  under  the  double  sign  of 
integration, 

0=zDj:  (cos.  y  D^  z)  +  Dy  (cos.  •/  D,j  z) 

=zcos.y{D%z+D'yz)+Dx  z  Dz  .  cos.  y-VDy  z  Dy  cos.  /.  (956) 

But 

Dz.  COS.  y=zDz.   {DzZ^  +  DyZ'^+X)"^ 

=  —  C0S.3  y  {DzzDlz  +  DyzDl.y  z),     (957) 

Dy.  cos.  Y  -^  — COS?  Y{DzzDl,yZ+DyzDlz)  ;     (958) 

which,  substituted  in  (956),  give  by  (952), 

^^  (1+D,  %^)  Dlz—2  DzzDyz  Dl.yZ  +  jX+D^z-^)  Dlz 

sec.3  y 

or       5'  =  — (?;  (960) 

so  that  this  surface  is  one  in  which  every  point  is  a  case 
of  <§)  158  ;  that  is,  i?i  which  the  curvatures,  in  directions 
'perpendicular  to  each  other ,  are  equal  and  opposite. 

The  plane  is  the  most  simple  instance  of  such  a  sur- 
face, but  there  are  other  examples  to  an  milimited  ex- 
tent. 

167.  Corollary.  The  complete  determination  of  these  sur- 
faces must  be  reserved  for  a  chapter  upon  the  integration  of 
partial  differential  equations  ;  but  the  following  ingenious  con- 
struction, proposed  by  Monge,  notwithstanding  its  obvious  want 
of  practical  utility,  which  was  acknowledged  by  its  author,  is 
17 


194  INTEGRAL    CALCULUS.  [b.  V.   CH.  VIII. 

Construction  of  minimum  surface. 

sufficient  to  exhibit  the  possibility  of  such  a  surface,  and  give 
some  idea  of  its  nature. 

Let  any  curve  line,  of  single  or  double  curvature,  be  drawn 
at  pleasure  in  space.  Produce  all  its  radii  of  curvature 
towards  the  opposite  side  of  the  curve  from  the  centres  of  cur- 
vature, and  to  a  distance  from  the  curve  exactly  equal  to  the 
corresponding  radii  of  curvature.  The  given  curve  line  may, 
then,  be  assumed  as  a  line  of  curvature  of  the  required  sur- 
face ;  that  is,  as  a  line  which  lies  upon  the  surface  and  has  at 
each  point,  the  same  curvature  with  the  surface  in  the  direction 
of  this  line.  The  produced  radii  of  curvature,  will  be  the  radii 
of  curvature  of  the  surface  in  directions  perpendicular  to  the 
given  curve  ;  and  if  the  extremities  of  those  produced  radii, 
which  are  the  corresponding  centres  of  curvature,  are  fixed, 
and  if  all  the  points  of  the  given  curve  are  rotated  with  the 
radii  about  these  centres,  moving  in  planes  perpendicular  to 
the  given  line,  each  element  of  the  given  line  will  describe  an 
element  of  the  required  surface.  The  given  line  in  its  new 
position  will  acquire  a  new  form  and  become  a  new  line  of  cur- 
vature, from  which  another  elementary  zone  of  the  surface  may 
be  described  by  a  repetition  of  the  above  process. 

The  small  arc,  through  which  each  point  of  the  curve  must 
move,  is  not  arbitrary,  but  is  limited  by  the  condition  that  two 
successive  radii  must  be  in  the  same  plane,  so  as  to  meet  at 
the  centre  of  curvature. 

168.  Corollary.  If  the  given  curve  of  the  preceding  con- 
struction were  a  circle,  the  resulting  surface  would  be  a  sur- 
face of  revolution  about  an  axis  perpendicular  to  the  plane  of 
the  circle  and  passing  through  its  centre.     The  particular  form 


<5>   168.]  CURVATURE    OF    SURFACES.  195 

Minimum  surface  of  revolution. 

of  this  surface  may  be  investigated  by  taking  the  axis  of  z  for 
that  of  revolution,  so  that  if 

,,  =:^2  +  y2^  (961) 

z  will  be  a  function  of  u^  and  will  contain  no  other  function  of 
X  and  7j.     Hence 

D^zz=zD^z.D^u  —  ^xD,z,  '] 

^  ^  '  K    (962) 

which,  substituted  in  (958),  give,  by  dividing  by  4  cos.3  y, 

2>„  2  +  2  ^^  B^z"^  ^uDlz=  0.  (963) 

By  putting  i;  =  a/  w,  (964) 


we  have  ^  1      r^ 

D^z  —  —-  D,z, 
Z  V 


B^  z— B  ^-1-  —~B^z' 


(965) 


which,  substituted  in  (963),  give 

B  z  A-  B  z^ 

—      [^  +D;z=  0.  (966) 

Hence  1  Biz         _ 

^  =  v+B^z+B^z^-^ 

\     .   Biz       B^zBlz 

the  integral  of  which  is,  by  introducing  A  as  an  arbitrary  con- 
stant, 

log.  ^=log.  I'+log.  B,  z— log.  V  (1+A  ^2),      (968) 


196  INTEGRAL    CALCULUS.  [b.  V.  CH.  VIII. 

Minimum  surface  of  revolution. 


or  A  D,.  z 


V        V{i+D,z2)' 


(969) 


Hence  _  A 

and  if  q  is  taken  so  that 

t;=^Cos.  y,  (971) 

(970)  gives 

Dip .  z=:  D^z  .  Dip  V  =1  A  Sin.  cp  D^z 

=  ^Sin.  ^  .  -r^^—   =  A,  (972) 

z=  Acp;  (973) 

and  the  equation  of  the  surface  is 


iAieUe-^)-  (^'^) 


§    169.]  CUBATURE    OF     SOLIDS.  197 

General  expression  for  the  element  of  volume. 


CHAPTER    IX. 


THE     CUBATURE    OF     SOLIDS. 


169.  Problem,     To  find  the  measure  of  the  volume 
of  a  given  solid. 

Solution.  Let  the  conditions  of  the  bounding  line  be  ex- 
pressed  by  an  equation  between  three  variables,  /,  m,  and  n. 
Suppose  two  surfaces  drawn  infinitely  near  each  other,  in  such 
a  way  that  n  is  constant  throughout  their  extent.  If,  then,  V 
denotes  the  required  volume,  we  have 

€?„  F=  the  lamina   included  between  these  two  surfaces. 

If  two  other  surfaces  are  drawn  infinitely  near  each  other,  in 
such  a  way  that  m  is  constant  through  their  extent,  we  have 

djn  dn  V=z  the  small  solid  rod  included  between  these  four 
surfaces. 

If  tw^o  more  surfaces  are  drawn  infinitely  near  each  other,  in 
such  a  way  that  /  is  constant  throughout  their  extent,  we  have 

di  djn  dn  V  =:  the  infinitely  small  parallelopiped  included  be- 
tween these  six  surfaces.  (9T5) 

If  s'  denotes  an  arc  of  the  intersection  of  two  surfaces  for 

which  7n  and  ?i  are  constant,  s"  an   arc  of  the  intersection  of 

two  surfaces  for  which  /  and  n  are  constant,  s'"  an  arc  of  the 

intersection  of  two  surfaces  for  which  /  and  /«  are  constant; 

17* 


198  INTEGRAL    CALCULUS.  [b.  V.  CH.  IX. 

General  expression  for  the  element  of  volume. 

and  if  a!  is  the  inclination  of  s"  to  s'"  at  the  point  of  meeting, 
a!'  that  of  s'  to  s'" ^  and  a!"  that  of  s'  to  s" ;  and  if  h  is  the  in- 
clination of  s'"  to  the  surface  which  includes  s'  and  5"  ;  the 
sides  of  the  small  parallelopiped  will  be   c?s',  ds" y  ds'"  ; 

the  face  which  includes  d  s'  and  d  s"^=  sin  a'"  d  s'  d  s" 
the  distance  of  this  from  the  opposite  face  =  sin.  h  ds'"  \ 

whence 

di  d^  d,  F=  sin.  a'"  sin.  h  ds'  d s"  ds'".         (976) 

But  I  is  the  only  variable  in  5',  m  the  only  one  in  5",  and  n  the 
only  one  in  s'" ,  whence  the  accents  may  be  neglected,  by  di- 
viding by  dl .  dm  .  dn  f  and  (976)  gives 

(977) 
Dl^,r^  V=D,,D^  .  Z>„  V-  sin.  a'"  sin.  b  D,  s  .  D,,s  .  D,,s  ; 

in  which  DiS,  D„^  s,  and  D„  s  may  be  deduced  from  the  gen- 
eral expression  for  the  differential  of  an  arc  in  space,  by  put- 
ting successively  each  pair  of  the  quantities  /,  m  and  w,  equal  to 
zero.     The  value  of  V  is,  then,  the  third  integral  of  (977). 

170.  Corollary.  If  one  of  the  vertices  of  the  parallelopiped 
is  taken  for  the  centre  of  a  sphere,  a\  a",  a'"  will  form,  by  the 
intersection  of  the  sides  of  the  parallelopiped  with  the  surface 
of  the  sphere,  a  spherical  triangle  ;  in  which  h  will  be  the  dis- 
tance of  a'"  from  the  opposite  vertex. 

Hence,  if  A'  is  the  angle  opposite  a',  and  if  M  is  the  ratio 
of  the  sines  of  the  sides  to  the  sines  of  the  opposite  angles,  so 

that 

sin.  A 
31=-. p,  978) 

sm.  a'  ^       ' 

we  have 

sin.  h  =r  sin.  a'  sin.  ^'=  iHf  sin.  a!  sin.  o!' ;       (979) 


§    173.]  CUBATURE    OF    SOLIDS.  199 


Cubature  of  solids  of  revolution. 


and  (977)  becomes 
Z>f.„.„.F=^-^sm.a'sin.  «"sin.  d^.D.s.D^s  .  D^s.  (980) 

171.  Corollary.  If  I,  m,  n  are  the  rectangular  coordinates 
z,  y,  z,  we  have  by  (725), 

ds^-  =  cl  x^  +  chf-  +  dz^  (981) 

a'  =  a"  =  a'"  =  ^^,     31  z=  i  ;  (982) 

and  (980)  gives 

Dl.,.^.V=l,  (983) 

V=f.fJ.  1  =U,  z  -/./.  y  =fyf.  ^.  (984) 

172.  Corollary.  If  /,  m,  n  are  the  polar  coordinates  of 
§  73  of  B.  I.,  the  equations  (31,  32,  33)  of  vol.  1  give,  by  put- 
ting 

u  :=.  r  sin.  ip  '\ 

y  z=i  u  cos.  6  >    (985) 

z  z=z  u  sin.  <5j  ) 

dy'^-\-dz^—d ifi  +  w2  ^^2  (9S6) 

=  f/r2+r2  rf52_|_^2sin.2^,rf^'2,  (987) 
D^  ^  ^F=r2  sin.  c/^,  (988) 

V=fl^^r^^^^n.,=-fl  ^  ^.2eos,,=/^^r3sin.,.(9S9) 

173.  Corollary.  If  the  coordinates  are  2:,  ?<,  5  of  the  pre- 
ceding corollary,  (987)  gives 

Dl.u.^V=u  (990) 


200  INTEGRAL    CALCULUS.  [b.  V.  CH.  IX. 

Volume  of  spliere,  ellipsoid. 

174.  Corollary.  If  the  given  solid  is  one  of  revolution  about 
the  axis  z,  of  which  a  segment  is  required  formed  by  two  planes 
perpendicular  to  the  axis  of  revolution,  z  may  be  substituted 
for  X  in  (991),  and  the  integrals  relative  to  6  taken  from  0  to 
2  -^.     Hence 

V—'XnJ^^     ^.u'=nf^.u'^  =  2nr.zu.       (992) 

175.    Examples. 

1.  To  find  the  volume  of  the  segment  of  a  sphere. 

Solution.  If  Jt  is  the  radius  of  the  sphere,  and  if  the  axis 
of  z  is  perpendicular  to  the  bases  of  the  segment,  (992)  gives 

=  ^RHz,-z,)  —  in(^zl^zl).  (993) 

Corollary.    The  solidity  of  the  sphere  is  fyr/ja^  (994) 

2.  Given  the  volume  Fof  a  solid  included  within  any  sur- 
faces whatever,  the  combination  of  which,  considered  as  one 
surface  which  in   general  is  discontinuous,  is  represented  by 

the  equation 

i^.  (a:  .  3/  .  z)  =  0,  (995) 

to  find  the  volume  V  of  a  solid  included  within  the  system  of 
surfaces 

^.(|>.l)  =  0.  (996) 

Ans.    V'=ahc  V.     (997) 

3.  To  find  the  volume  of  the  segment  of  an  ellipsoid  in- 
cluded between  two  planes  drawn  perpendicular  to  either  of 
the  axes  of  the  ellipsoid. 


«§>  175.]  CUBATURE    OF    SOLIDS.  201 


Volume  of  hyperboloid  and  paraboloid. 


Ans.  U  A,  By  C  are  the  axes  of  the  ellipsoid,  if  the  planes 
are  drawn  perpendicular  to  the  axis  of  C,  and  if  V  is  the 
solidity  of  the  segment  of  a  sphere  whose  radius  is  unity,  the 
segment  being  included  between  two  planes  drawn  at  the   dis- 

tances -^  and -^  from  the  centre,  the  required  volume  is 

F'=  ABCV.  (998) 

4.  To  find  the  volume  of  the  segment  of  an  hyperboloid  in- 
cluded between  two  planes  drawn  perpendicular  to  that  axis, 
for  which  the  sections  made  by  the  planes  are  elliptical. 

Ans.  If  Cis  the  axis  perpendicular  to  the  planes,  and  if  ^ 
and  B  are  the  other  two  axes,  the  required  volume  is 

V=i^^  {zl-zl)  ^nAB  {z-z,),      (999) 

in  which  the  upper  sign  corresponds  to  the  hyperboloid  of  one 
branch,  and  the  lower  sign  to  the  hyperboloid  of  two  branches. 

5.  To  find  the  volume  of  the  segment  of  the  paraboloid,  in- 
cluded between  two  planes  drawn  perpendicular  to  the  axis  of 
z,  the  equation  of  the  paraboloid  being 

(-.)■+ (!)'=  <^-  <■■> 

Ans.     ^n  ABC{z\  —  zl).         (2  a) 

6.  To  find  the  volume  of  the  segment  of  a  solid  of  revolu- 
tion included  between  two  planes,  drawn  perpendicular  to  the 
axis  of  revolution,  when  the  revolving  arc  is  that  of  a  cycloid 
about  the  axis  of  x  in  (130)  of  vol.  1. 

Ans,     Vz=i  I  jR2  jt  (sin.  2  fl,— sin.  2  ^^o)— 2  R-  ^  (sin.  ^,— sin.  ^o) 
+  3i22-(^-^o).  (3a) 


202  INTEGRAL    CALCULUS.  [b.  V.   CH.   IX. 

Solid  of  least  surface. 

7.  To  find  the  volume  of  the  segment  of  the  solid  of  revo- 
lution of  §  174,  when 

u  =  B  Cos.  J.  (4  a) 

Aus,  V=iAB'-n(Sm!^-^'-Shi^-^\-\-iB^7v{z^-z,).  (5a) 

176.  Prohlein.  To  find  the  inaximuin  or  ininimiini 
volume  which  can  he  included  by  a  surface  drawn  under 
given  conditions. 

Solution.  Since  the  general  expression  for  the  volume  is 
reduced  to  the  form  of  a  double  integral,  this  problem  is  pre- 
cisely similar  in  its  solution  to  that  of  §  165. 


177.     Examples. 

1.  To  find  the  maximum  or  minimum  volume^  which 
can  be  included  within  a  surface  of  a  given  area. 

Solution.  Since  the  double  integral  (984)  is  to  be  a  maxi- 
mum, while  that  of  (811)  is  to  be  constant.  We  have,  by 
§  166,  if  ^  is  a  constant  multiplier, 


or  1    I   1  ^ 


+  '-  =  -i;  (7  a) 


that  is,  the  surf  ace  is  one  for  which  the  sum  of  the  recip- 
rocals of  the  greatest  and  least  radii  of  curvature  at  each 


<5>    177.]  CUBATURE    OF    SOLIDS.  203 

Solid  of  revolution  of  least  surface. 

point  is  constant.  The  general  equation  of  this  sur- 
face has  never  been  obtained,  but  the  sphere  and  the 
cylinder  are  evidently  cases  of  it. 

2.  To  find  the  solids  of  revolution  which  are  solutions  of  the 
preceding  problem. 

Solution.    Let  the   axis  of  z  be  that  of  revolution,  and  by 

putting 

u  =  ^(x^+y^),  (8  a) 

(7  a)  becomes,  by  means  of  (952), 


Let  V  be  taken  so  that 

uD, 

V 


(10  a) 


whence 

log.  V  =  log.  u  +  log.  D,,z  —  i  log.  (I  +  Z),  z2)  ;      ( 11  a) 

the  differential  of  which  is 

D^v       I    .  Dlz        D^z  Dl  z 

—  =--rTr 


V      ~~  u    '  D,z  l+D^z^ 

1.  ,     mz 

u  '^  Dl{\+D,,z') 


=  ^+7^o4rV7r^;  (12a) 


which,  multiplied  by  (10  a),  gives  by  (9  a), 

(13  a) 

The  integral  of  this  equation  is 


u-' 


v  =  -^-^  +  B,  (14a) 


204  INTEGRAL    CALCULUS.  [b.   V.   CH.   IX. 

Solid  of  revolution  of  least  surface. 

in  which  JB  is  an  arbitrary  constant.     But  if  r  is  taken  so  that 

D„  z  :=  cot.  T,  (15  a 

(10  a  and  14  a)  give 

V  =:  u  COS.  T,  (16  a 

COS.r=:  — -^  +  -   ,  (Ha 

2  A         u 

V(-^  +  cos.3^)=-+-,  (18a 

u  =Acos.r-/y{2AB+A''cos.\),  (19  a 

^1       ^^sin.  tCOs.  T 

^  '  /s/(2AB+A^cos.Ty  ^ 

-A  ^  cos  ^  "^ 

^  '  //(2^S+^^cos.^t)    ^ 

If  e  is  taken  so  that 

'=    ^{A2  +  2ABY'  (^^^ 

(21  a)  gives,  by  the  notation  of  elliptic  integrals, 

\/(2  A  B+A^  cos, 2  r)—  a/(^2_j_2  AB)  ,^r,     (23  a 
,      A^—2AB         ,  2^^ 

i>^.=.-^  COS.  ^+;7^^  V,  ^^^, ,  x+ V(Z^^-^)^^-  (''' 

z  =  —  ^  sin.  T  -|-  (yl_2  B)eF'^  +  2eBEr  ;  (25  a 

and  z  may  be  found  in  in  terms  of  u,  by  substituting  (19  a)  in 
(25  a). 

The  preceding  solution  applies  strictly  to  that  case  only  in 
which  A  and  B  have  the  same  sign  ;  for,  when  they  have  op- 
posite signs,  e  becomes  greater  than  unity,  and  when  B  is  also 
greater  than  A,  e  is  imaginary ;  but  these  cases  are  solved 
without  difficulty. 


<§»   177.]  CUBATURE    OF    SOLIDS.  205 

Greatest  solid  under  given  conditions. 

3.  To  find  the  greatest  solid  of  all  those  for  which 

/./.sec.2y,  (26  a) 

has  a  given  value,  7  being  the  inclination  of  the  tangent  plane 
of  the  bounding  surface  to  the  given  plane  of  2?/. 

Solution.    If  ^  is  the  constant  multiplier  of  (26  a),  the  equa- 
tion of  the  maximum  is 

l  —  A-'Dlz  —  A-'D;z=0,  (27  a) 

which  is  easily  derived  from  the  equation 

sec.2  v  =  l  +  D,  z"^  +  /),  z\  (28  a) 

Let  V  be  taken  so  that 

z=:lA{x  +  yYJ^v,  (29a) 

which  gives 

Dlz^^A+Dlv,  (30  a) 

Dl%  =  iA  +  Dlv-  (31a) 

which,  substituted  in  (27  a),  give 

Dlv+Dlvz=:0.  (32  a) 

Let  now 

mz^  x-\-  y  \/ —  1 ,       w  —  X  —  y  s/ —  1  ;  (33  a) 

and  w^e  have 

DyV  ziz  {D„,v  —  D,v)\/\,  W34a) 

Dlv  =  Dlv\-<^  Dl,„v  +  Dlv, 
/>>  =  — Z>>  +  2  J9L.  V  —  Dl  V  ;   J 

which,  substituted  in  (32  a),  give 

r  0     —Dm.D,  V.  (35  a) 


>2 

m  .  n 


18 


206  INTEGRAL    CALCULUS.  [b.  V.  CH.  IX, 

Greatest  solid  under  given  conditions. 

Hence  D„  i?  is  a  function,  whose  difTerential  coefficient  taken 
relatively  to  m  is  constant,  and  may,  therefore,  be  any  function 
whatever  of  n,  represented  by  N ;  that  is, 

Dr.  V  =  N,  (36  a) 

Hence  v  =f,  N —f.71  +  F.  nij  (37  a) 

in  whichy  and  JP  are  any  arbitrary  functions  ;  f .  n  is  the 
function  whose  differential  coefficient  is  N,  and  F.ni  is  the 
arbitrary  quantity  which  is  constant  relatively  to  n ;  that  is, 
which  does  not  vary  with  n,  but  may  be  any  function  whatever 
of  the  other  variable  m,  and  which  is  added  to  complete  the 
integral.     By  the  substitution  of  (33  a),  (37  a)  gives 

v=f.{x  +  ys/-\)+F[x-y^^\).         (38  a) 

If  we  put  F  —  f'^^—i,Fi  _     (39  a) 

F^f'--    ^—l,F\  (40  a) 

in  which  f  and  F'  are  real  functions,  the  value  of  y  becomes 

(41  a) 
»=/.(x+V-l)+/.(a:-^-l)+v'-l[-F'(^+'«/-l)--F'(^v-I)], 

from  which  the  imaginary  quantities  will  wholly  disappear. 


<§>    180.]  LINEAR    DIFFERENTIAL  EQUATIONS.  207 

Order  and  degree  of  differential  equations. 


CHAPTER   X. 

INTEGRATION  OF  LINEAR  DIFFERENTIAL  EQUATIONS. 

178.  A  differential  equation  is  said  to  be  of  the  same 
oi'de?'  with  that  of  the  highest  differential  coefficient 
which  it  involves. 

The  degree  of  a  differential  equation  is  determined 
in  the  same  way  as  that  of  an  ordinary  equation,  except 
that  the  independent  variables  are  neglected,  and  each 
differential  coefficient  is  counted  as  a  variable. 

Thus  the  equation 

A  Dlv  +  B D"-^ V  +  &z.c.-\-  A'Dlv  -\-  B' D-'  v  +  &c. 

-\-E  Dl-"".  D';;  V  4-  &c.  +ez;  -{-n=0.  (42a) 

is  of  the  n  order  ;  but  it  is  only  of  the  first  degree,  or  linear, 
if  the  coefficients  A,  B,  &,c.  involve  the  independent  variables 
X,  y,  &:,c.,  but  do  not  involve  v,  &/C. 

179.  Any  equation,  which  is  of  a  less  order  than  a 
given  differential  equation,  and  satisfies  it  by  the  aid 
of  differentiation  without  the  assistance  of  any  other 
equation,  is  said  to  be  an  integral  of  the  given  equation. 
The  integral  is  said  to  be  complete  when  it  contains  the 
greatest  possible  number  of  arbitrary  quantities. 

180.  Problem.  To  integrate  several  given  equations^ 
between  the  variables  x,  y,  z,  i^'c,  and  their  differential 
coefficients  taken  with  respect  to  the  independent  varia- 


208  INTEGRAL    CALCULUS.  [b.   V.   CH.   X. 


Linear  diHerential  equations  with  constant  coeliicienls. 

hie  tj  xolien  the  given  equations  are  linear^  and  contain 
no  term  independent  of  x,  y,  z,  t^'c,  and  wJien  all  the 
coefficients  are  constant,  and  the  number  of  equations 
the  same  with  that  of  x,  y,  z,  ^*c. 

Solution.     If  the  following  expressions  are  assumed  for  the 
variables 

x  =  A  c'\     y  —  Be''  &c.,  (43  a) 

in  which  5,  A,  B,  &/C.,  are  constant,  their  differentials  give 
DtX=i  As  e'\     Dty—  Bs  e'',  &lc.  ) 

D]x=As^e"-,   D\y  —  Bs^e'\^(i.  C    (44a) 

&c.  &c.  3 

If  these  values  are  substituted  in  the  given  equations,  these 
equations  will  evidently  become  divisible  bye*';  and  the  di- 
vision by  this  factor  will  free  the  equations  wholly  from 
variables,  and  reduce  them  to  equations  between  s,  A^B,  &c,, 
in  which  A  and  B  will  have  a  linear  form.  If  all  of  the  con- 
stants A^  B,  &LC.  but  one,  as  A,  are  eliminated,  the  result  will 
be  a  single  equation  involving  A  and  s,  in  which  A,  however, 
will  be  a  factor  of  the  whole  equation  ;  so  that  the  division  of 
this  equation  by  A,  will  lead  to  a  final  equation,  involving  no 
other  unknown  quantity  but  s,  and  which  will  serve  to  deter- 
mine s.     Let  the  equation  for  determining  5  be  denoted  by 

^  =  0,  (45  a) 

and  each  root  of  it  will  give  corresponding  values  of  A,  B, 
&/C.,  or  rather  of  their  ratios,  and  thence  values  of  x,  y,  &>c., 
which  will  be  integrals  of  the  given  equations. 

181.  Corollary.  The  number  of  integrals  found  by  the 
preceding  process,  will  be  the  same  as  that  of  the  different 
roots  of  the  equation  (45  a) ;  but  all   these  integrals  can  be 


<5»   183.]  LINEAR    DIFFERENTIAL    EQUATIONS.  209 

Linear  differential  equations  with  constant  coefficients. 

united  into  one  expression.  For  it  is  evident  that,  if  x^ ,  y^ , 
&c.  denote  any  one  of  these  systems  of  integrals, 

x^z  Lx,-\-L' x,,-\-^z.y     y=^Ly^-[- L' Us' -{•  ^C"     (46  a) 

will  also  be  a  system  of  integrals,  in  which  L,  i',  6lq,.  will 
be  arbitrary  ;  for  the  linear  form  of  the  given  equations  will 
cause  the  multipliers  of  L,  L\  &,c.  to  become  th  esame  func- 
tions of  x^ ,  y^ ,  &,c.,  which  the  whole  equations  are  of  x,  y, 
&,c.  ;  and  therefore  x,, ,  y^ ,  &/C,  will  satisfy  the  equations  in 
the  same  way  as  they  do  when  they  are  by  themselves  ;  that  is, 
the  aggregate  of  the  terms  dependent  upon  them  will  be  zero. 

182.  Corollary.    If  the  first  member  of  the  equation  (45  a) 

is  reduced  to  the  form 

s"  +  «  s"-^  +  &c.,  (47  a) 

the  expressions 

will,  by  the  notation  of  the  residual  calculus,  include  all  the 
terms  of  (46  a),  provided  that  the  residuation  is  performed 
relatively  to  s,  and  that  Aj  B,  C,  &lc.  assume  a  new  system  of 
values  for  each  root  of  <S^.  The  forms  (48  a)  might,  indeed,  be 
directly  applied  to  the  integration  of  the  equation,  by  perform- 
ing the  differentiation  under  the  sign  of  residuation. 

183.  Corollary.  It  may  be  remarked,  that  A,  B,  &lc.  are 
integral  polynomials  in  terms  of  s  ;  and  if,  indeed,  they  were 
not  so,  the  multiplication  of  each  of  them  by  their  common 
denominator  would  reduce  them  to  such  polynomials.  Neither 
of  them  is  a  polynomial  of  a  higher  degree  than  the  (« — l)st  ; 
for  if  either  of  them  were  of  a  higher  degree,  the  division  by  iS^ 
would  reduce  such  a  term  to  the  form 

QS+R,  (49  a) 

18* 


210 


INTEGRAL    CALCULUS. 


[b.   V.  CH.  X. 


Linear  difTerential  equations  with  constant  coefficients. 

in  which  R,  the  remainder  of  the  division,  must  be  of  a  less 
degree  than  S ;  and  (49  a)  is  by  (45  a)  reduced  to  R. 

184.  Corollanj.  If  the  values  of  A,  B,  &c.,  in  (48  a), 
change  their  values  for  different  roots  of  S,  only  so  far  as  they 
should  to  conform  to  the  sign  of  residuation,  the  values  (48  a) 
will  involve  only  one  arbitrary  constant,  which  is  a  factor  of 
each  of  the  quantities  A,  B,  &/C.  But  if  this  arbitrary  con- 
stant is  reduced  to  unity  ;  and  if^,  B,  &lc.  are  then  multiplied 

by  the  polynomial 

«5«-i_j_^^^."-2_j_&,c.,  (50  a) 

iji  which  «,  (^,  &:.c.  are  arbitrary  constants,  the  requisite  num- 
ber of  arbitrary  constants  is  again  introduced  into  (48  a).  The 
values  of  (48  a)  may,  by  the  process  of  §  183,  be  reduced  to 

the  form 

(51a) 
_  f  {L,  a+M,  ?+&c.)e"     _  ^  {L,a+M,J-\-&i.c.y 

((«)) 


C 


((«)) 


y=L 


,  &c. 


in    which    L^ ,  Ly,  31^ ,  3Iy ,    &c.    are    integral   polynomial 
functions  of  s,  neither  of  which  exceeds  the  (n — l)st  degree. 


185.  Corollary.    The  dilferentials  of  (51  a)  become,  by  the 
same  method  of  reduction, 

(Lx  «  +  Mx?  -\-  &c.)  se'' 


D,x- 


{{S)) 


-  C  ((^)) 


D]x  =  l 


{L:a-{-M:?+&LZ.)e' 


({S)) 


j^tv  —  c  ^i^s)) 


&c. 


>  (-52  a) 


<§>   186.]  LINEAR    DIFFERENTIAL    EQUATIONS.  211 

Integration  of  linear  differential  equations  with  constant  coefficients. 

186.  Corollary.  If  x^  ,  ij  ^  ,  (fcc.  ;  Jo,  ?/o ,  &lc.  ;  z'J ,  &/C., 
represent  the  values  of  cT,  y,  &/C. ;  D^  x,  D^  y,  &/C.  ;  D]x^  &/C. 
when  ^  vanishes;  and  if 


/ 


(53  a) 


equations  (51a  and  52  a)  give 

If  the  number  of  the  equations  (54  a)  is  taken  equal  to  that  of 
the  constant  «,  ,'?,  &c.,  the  values  of  «,  (5,  &,c.,  may,  by  the 
usual  process  of  elimination,  be  found  in  terms  of  Xq  ,  i/^  ,  &:,c. 
The  expressions  of  «,  ^,  &;c.  in  terms  of  x^  ,  i/^  ,  &;c.  will 
clearly  be  linear  functions  of  «,  (^,  &c.  ;  so  that  if  these  values 
are  substituted  in  (51  a  and  52  a),  the  expressions  of  t,  y,  &/C. 
will  contain  x^ ,  y^  ,  &c.,  in  the  same  linear  form  in  which 
they  now  contain  «,  ,^,  &/C.  The  values  of  «,  i^,  &lc.,  in  (51  a 
and  52  a),  might,  then,  have  been  assumed  at  once  as  identical 
with  Xq  ,  ?/q  ,  &c.,  and  the  corresponding  values  of  x,  ?/,  &c. 
would  be 

J.  Zz  a;^  +  Z;  x'o  +  &c.  +  iJ/x  1/q  +  &c. 


X  =z  l^ " —     . ,  ^.  X — ^ e' 


&LC,. ; 


(55  a) 


s£ 


2'  -  C  ((Sj) 


in  which,  it  may  be  observed,  that  the  values  of  Z/j,  Mx,  &/C. 
are  entirely  distinct  from  those  in  (51a  and  52  a). 


212  INTEGRAL    CALCULUS.  [b.  V.  CH.  X. 

Residual  integral  of  a  rational  fraction. 

187.  Lemma.    If  JP  denotes  the  value  which  xf.x 
acquires  when  x  becomes  infinite,  we  have 

F=l{(f.x)),  (56  a) 

whenever  /.  x  denotes  a  rational  fraction,  of  which  the 
degree  of  the  numerator  is  less  than  that  of  the  de- 
nominator. 

Proof.     It  follows  from  (216  and  219),  that,  in  the  present 


case. 


the  product  of  which  by  x  is 

.f..  =  lp^\  ,  (58  a) 

But  when  x  is  infinite,  (58  a)  becomes 

F=l{{f.z))=t{{f.^)).  (59  a) 

188.  Corollary.  When  the  excess  of  the  degree  of  the  de- 
nominator of /.  x  above  the  numerator  is  greater  than  unity, 
(59  a)  becomes 

0=t((/-^)).  (60  a) 

189.  Corollary.  When  the  excess  of  the  degree  of  the  de- 
nominator of y.  X  above  the  numerator  is  exactly  unity,  and 
when  f  ,x  is  of  the  value  (217),  (59  a)  becomes 

a 
'a! 


-,^L{U'A)'  (61a) 


190.  Corollary.    Since,  when  t  becomes  zero,  the  values  of 
X,  y,  &/C.  (55  a)   are  reduced  to  x^  ,  y^  ,  &,c. ;  the  polyno- 


<§)  190.]  LINEAR    DIFFERENTIAL    EQUATIONS.  213 

Integration  of  linear  differential  equations  with  constant  coefficients. 

mials  L^ ,  31^  ,  Ly  ,  L'y ,  &c.  must  be  of  a  less  degree  than 
the  (n —  l)th;  while  Lx  ,  My  ,  &c.  must  be  of  the  form 

5"-i  4- J  s"-2  +  &c.  (G2a) 

The  form  of  5  L^  is,  therefore, 

s"  +  b  s"-' -{- &LC.  ;  (63  a) 

so  that,  by  (47  a),  s  i^  —  S  (64  a) 

is  of  a  less  degree  than  S.    We  have,  then,  by  denoting  (64  a) 

But  when  t  vanishes,  Dt  x  is  reduced  to  Jq  >  and  therefore 
s  Lt'x  must  be  of  the  form  (62  a),  while  L,^, ,  s  Mz  ,  &lc.  must 
be  of  a  smaller  degree.  We  have  then,  again,  by  the  differen- 
tiation of  (65  a), 

and  a  similar  train  of  argument  may  be  continued  to  the  higher 
differential  coefficients. 

191.  Corollary.  If,  in  the  given  equations,  there  are 
substituted  for  x,  D^  x,  D]  x,  &,c.,  the  quantities  contained 
under  the  sign  of  residuation  in  (55  a,  Go  a,  6G  a,  &lc.),  those 
equations  must  be  satisfied.  The  reverse  process,  therefore, 
of  substituting   for  DtX,  D^y,  &:-c.,   not  5  z,   sy,  &,c.,   but 

sx  —  2;q>S^.— ,     sy — .yo^-^,    <Scc.,     and   for   D'^x,   &:,c., 

s^  X  —  {x'o-\-sxA  S  —  ,  &LC.,  must  give   again  the  parts  of 

o 

X,  y,  &/C.  in  (55  a),  which  are  under  the  sign  of  residuation. 


214  INTEGRAL    CALCULUS.     •  [b.  V.   CH.  X. 

Integration  of  linear  difTerential  equations  with  constant  coefficients. 

192.  Corollary.  If  ^  be  taken  to  denote  the  expression  for 
S  when  D^  is  substituted  for  5,  and  if  Jt,  ^H,  &c.,  denote  the 
expressions  whi6h  L,  3J,&lc.  assume  by  the  same  substitution, 
and  if  ®  be  taken  so  that 

,     •        -  =  im)'  ^'''^ 

we  shall  have  «Z/e*'  .^^    ^ 

^pypSO,  (68  a) 

and  the  values  of  x,  y,  &.c.  (55  a)  will  become 

y=(2L,:r,+iL>-;  +  &c.+  JH,yo+&'C.)6),       5      ^'^'^ 

in  which  |iz  &.C.  are  not  proper  factors,  but  express  functional 
operations  to  be  performed. 

The  value  of  r  would  be  obtained  by  eliminating  x,  1/,  &,c. 
directly  from  the  given  equation,  in  which  process  D^^D], 
&c.  are  to  be  treated  as  though  they  were  factors.  The  values 
of  X,  y,  &/C.  (70  a),  will  then  be  obtained  by  the  same  process 
of  elimination,  from  the  equations,  which  are  obtained  from 
the  given  equations,  by  substituting 

DtX  —  a^o  ^  ®         for  DiX,  ^ 

Ay  — yo^®        forAy,  ifcc.  >    (^la) 

Djx  — (xj  +  x^  D)  re,  ) 

&.C. 

This  is  Cauchy's  method  of  integration,  and  the  function  ®  is 
called  the  principal  function. 


<5>    194.]  LINEAR    DIFFERENTIAL    EQ,UATIONS.  215 

Linear  differential  equations  with  constant  coefficients. 


193.  Corollary.  When  the  equation  (45  a)  has  several 
equal  roots,  the  corresponding  systems  of  values  in  (46  a) 
would  seem  to  coalesce  into  one.  This  loss  of  terms,  and 
therefore  of  arbitrary  constants,  is,  however,  unnecessary  ;  for 
if  the  roots  5,  s',  s",  &c.,  instead  of  being  equal,  differed  infi- 
nitely little  from  each  other,  so  that 

s'=s  +  A,  s"=s'+h'z=is+li+h',  &c.,  (72a) 
in  which  h,  h\  &c.  are  infinitely  small,  we  shall  have,  by 
(416)  of  vol.  1,  upon  putting 

Bz^A'h,    B'z=A"h\     C  =  B'h,  &c.,  (73  a) 

>-(74a) 

A"e'"'=A"e'''+B't  e'''=A"e''+{A"h-\-B')t  e''-hC t^  e'') 

&c. 
The  new  terms,  multiplied  by  t,  t'^,  &lc.,  which  are  thus  in- 
troduced, are  just  sufficient  to  replace  those  which  are  lost  by 
addition.  These  very  terms  are  also  introduced  by  the  process 
of  residuation,  for  this  process  requires,  by  (182),  one  or  more 
diflferentiations,  whenever  the  roots  are  equal,  and  each  dif- 
ferentiation will  have  to  be   applied  to  e'^  in  (55  a).     But  by 

(481)  of  vol.  1, 

DT  e'^zzzt"^  e\  (75  a) 

whence  the  differentiation  will,  evidently,  introduce   the  re- 
quired terms. 

194.  Problem.  To  integrate  several  linear  differen- 
tial equations  between  the  variables  x^  y,  4*^.,  and  their 
differential  coefficients  taken  relatively  to  the  independent 
variable  t,  lohen  all  the  coefficients  are  constant^  the 
tei^ms  lohich  are  independent  of  x,  y,  cj'c.  are  given 
functions  of  t,  and  the  number  of  the  equations  is  the 
same  with  that  of  the  variables. 


216  INTEGRAL    CALCULUS-,  [b.  V.   CH.  X. 

Linear  differential  equations  with  constant  coefficients. 

Solution.  When  llie  functions  of  t  are  reduced  to  zero, 
this  problem  coincides  with  tlie  preceding  one  ;  and  if  ^,  v, 
&LC.  denote  the  corresponding  vahies  (70  a)  of  x,  y,  &c. 
obtained  by  the  preceding  process  ;  while  X,  Y,  &c.  are  par- 
ticular values  of  x,  ?/,  &,c.,  which  satisfy  the  present  problem, 
the  values 

x  =  t-\-  X,    y  =  r,+  Y,  &,c.  (76  a) 

are  complete  values  of  x,  y,  &,c.  for  £,  t],  &/C.,  involve  the  re- 
quired number  of  arbitrary  constants. 

The  problem  is  reduced,  then,  to  obtaining  these  particular 
values  of  x,  ?/,  &/C.  For  this  purpose,  let  the  subsidiary 
quantity  t  be  introduced,  and  let 

%=^-'^e=t^l,  (77  a) 

so  that  0  is  the  value  which  0  assumes  when  t  is  changed  into 
t  —  T.  If,  then,  X,  IT,  ^-c.  are  the  values,  which  i,  v,  &lc. 
assume,  when  ©  is  changed  to  0  ,  and  when  for  a-^ ,  x'q,  yo, 
&c.  are  substituted  ^^  ,  ^^ ,  ^^  ,  &,c.,  which  are  functions 
of  Tj  and  if  the  integrations  in  the  following  formulas  are  per- 
formed relatively  to  t,  we  may  put 

X  =  fl,X,     Y  =fi  Y,  &c.  (78  a) 

The  differentiation  of  (78  a)  relatively  to  t,  involves  not  only 
the  differentiation  of  X,  Y,  &c.,  under  the  sign  of  integration, 
but  also  the  changes  arising  from  the  change  in  the  limits  of 
integration.  If  then  X^,  '^t,  ^c.,  are  the  values  which 
v\,  "ST,  fcc.,  assume  when  t  is  changed  to  t,  the  differentiation 
of  (78  a)  gives 

D,X=X,+f,D,x,  •. 

D,Y=,  ¥,  +fi  D,  ¥,  &c.  5  (^^^^ 


<§)   194.]  LINEAR    DIFFERENTIAL    EQUATIONS.  217 

Linear  differential  equations  with  constant  coefficients. 

If,  again,  we  put 

X'z=D,x,     ^'  =  D,Y,  (80  a) 

another  differentiation  of  (79  a)  gives 

D',JC=zD,  X,  +  X]  +f'o  D]  X,  &c.  (81  a) 

By  the  substitution  of  X,  F,  &/C.  for  x,  y,  &c.,  in  the  given 
equations,  the  terms  under  the  sign  of  integration  must  dis- 
appear, for  the  terms  under  this  sign  in  the  values  of  JT,  D^  F, 
&,c.  differ  from  the  values  of  t,  v,  &c.  in  nothing  but  the  com- 
mon factor  e~^^,  and  the  writing  of  the  particular  forms  ^j, 
STi ,  &c.  for  the  arbitrary  constants  a;^  ,  Xq,  &;c. 

The  substitution  of  t  for  t  reduces  t^^^—'^)  to  unity,  and  if 
^z  ?  T'x  y  are  the  same  functions  of  t,  which  JTx  and  ^T^  are  of  t, 
we  have,  by  §  190, 

___  »  x^  r.  +  l:  t: + &C.+M,  7;+&c.  _ 

_  ^.Zx'  r.  +s  xj  r;+&c.+5irf,  7;+&c._^,       p    ^^ 
^-L  ((^))  ^-     -I 

T,  =  7;,   &c.  ^ 

Hence,  by  the  omission  of  the  parts  of  (78  a,  79  a  and  81  a), 
which  are  under  the  signs  of  integration,  they  become 

jr=0,     D,  X=  n  ,     D]  X—  %-\-D, .  Tx  ,  &c.  ^ 

and  the  substitution  of  these  values  in  the  given  equations, 
reduces  them  to  linear  differential  equations  in  which  T^c  ,  T^  , 
Ty  are  the  variables,  and  the  order  of  the  equations  is  less  by 
one  than  that  of  the  given  equations.  Thus  the  number  of 
these  variables  being  greater  than  that  of  the  equations, 
19 


218  INTEGRAL    CALCULUS.  [b.  V.  CH.  X. 

Linear  differential  equations  with  constant  coefficients. 

enables  us  to  take  certain  of  them  at  pleasure.  Thus  of  the 
quantities  Tx  ,  T^ ,  &c.,  all  but  one  may  be  supposed  to  be 
zero  ;  of  Ty,  T^ ,  all  but  one  may  be  zero  ;  and  in  the  same 
way  with  the  others. 

The  selection  of  the  quantities  T^ ,  &c.,  which  are  to  re- 
main of  a  finite  value,  is  immediately  fixed,  by  the  considera- 
tion that  the  resulting  equation  should  be  of  as  low  an  order 
as  possible.  It  is  generally  possible  to  select  those  quantities 
which  correspond,  respectively,  to  the  highest  order  of  diffe- 
rential coefficients  of  a;,  i/,  &c. ;  and  with  this  selection  the 
resulting  equations  are  wholly  free  from  difTerentials,  and  are 
solved  by  simple  elimination.  In  any  case,  however,  it  seems 
possible  to  make  a  selection  which  will  avoid  the  necessity  of 
integration. 

195.  Corollary.  When  s  is  nothing,  the  values  of  X,  Y 
must  vanish,  as  well  as  all  their  diflferential  coefficients  of  an 
order  inferior  to  those  which  correspond  to  the  quantities  in  the 
series  T^^  Ty,  &c.,  which  are  retained  as  finite.  Hence  the 
corresponding  values  of  x,7/,  D^x^  &c.  will  be  reduced  to 
^^o  >  yo>  ^0,  &c. 

196.     Examples. 


1.  To  integrate  the  diflferential  equation 

D'^x  +  a  D"r'  X  +  &c.  =  C/;  (84  a) 

in  which  U  is  a.  function  of  t. 

Solution.    In  this  case,  the  value  of  r  becomes  at  once 

F  =zD'l  +  a  Dr'  +  &^o.     z=f.Dr,  (85  a) 


<5>   196.]  LINEAR    DIFFERENTIAL    EQUATIONS.  219 

Linear  differential  equations  witli  constant  coefficients. 

Hence  S  =z  s""  +  a  s""^  +  &c.      =/.  s,  (86  a) 

by  taking/",  to  denote  the  integral  function,  which  constitutes 
the  second  members  of  (85  a  and  86  a).     We  have  also 

and  the  equation  for  determining  ^  is,  by  (71  a), 
(2>r  +  aDr'  +  &c.)^ 

-[xo(i>r'+a^r'+&c.)+x;  (z>^^+az>r'+&c.)+<^c.]/7  0=0 

or  (88  a) 

r  ^-[a:o(Z)r'+a^r'+&c.)+2;;(J9r2+aZ)r'+&c.)+&c.]f  0=0  ; 

whence  (89  a) 

If,  in  the  development  of  the  expression 

the  exponents  n,  n — 1,  &c.  of  x^  ,  are  regarded  as  expressing 
the  number  of  accents,  and  if  the  term  which  does  not  contain 
Xq  is  multiplied  by  x^  the  value  (89  a)  may  be  expressed  in  a 
more  simple  form ;  for  we  shall  have 

J^t  —  ^o 

To   obtain  the  value  of  X,  let  Wi  be  the  value  which   U 

assumes  when  t  is  changed  to  t,  and  by  omitting  the  accent  of 

Tx  as  unnecessary,   we  have  by  §  194,   if  we  suppose  all  the 

quantities  in  the  series  Tx  ,  &;c.  to  vanish  but  the  (n — l)st, 

T,=U,    K.  =  m;  (92  a) 


220  INTEGRAL    CALCULUS.  [b.   V.  OH.  X. 

Linear  differential  equations  with  constant  coefficients. 

whence,  by  (78a,  76a  and  Ola), 

X=fl,m0^  (93  a) 

_/-^^-/^o0_j_y..2g0  (94a) 


a;  =  — 


O.-^o 


2.    To  integrate  the  differential  equation 

Solution.    In  this  case,  we  have 

f.D,  =  D]—{a  +  b)D',  +  abD,  =  D,{D,  —  a)  {D,  —  b) 
S=  s  (s  — a)(s  — 6) 

i^^^  =x^[D]-^(a+b)  A+a  b]+x',  [l>,-(a+6)]+x,' 

^_f'Dt-f'^o  y  x^[s2-(a-^b)s+a  b]-{-x'o  [s-{a+b)]+x'^ 

'-    A-^o        ^^  ({s(s-a){s-b))) 

__  ct  +  b      ,    ,      1        „ 

—  •*'o  """  I         ■*'0     t  I.     ^0 


^''  +  rr—ix  ^  > 


a  (a — b)  b  (« — 6) 


T> 


^-y  0  T-C((5(5_^)(5_^,)  ))—  C52  ((5(s— «)   (S— 6)  )) 

_  ct^       c  e""'— cjat  +  1)        ce''  —  c(bt  +  l) 
~  2Vb~^  ^(a—b)  63  (^a—b) 

and       z  =:  I  -f-  JT. 

3.    To  integrate  the  differential  equation 

D',  X  —  S  a  D]x  +  S  a^  D.x--  a^  X  =  b  c^ 


<5>   196.]  LINEAR    DIFFERENTIAL    EQ,UATIONS.  221 

Linear  difierential  equations  with  constant  coefficients. 

Solution.    In  this  case,  we  have 

S=  (s— a)3 
'^'^''^''''=x^(D-a)^+{x',-ax^){D,-a)+{x'--2axl+a^x,) 

!>  Xq  (s—aY+{xo—a  Xp)  (s—a)+{x'^--2  a  x'^+a^  x^) 

4.   To  integrate  the  diflferential  equation 

D\x  -\-  a^  DtX  ■=.})  sin.  m  t. 

Solution.    In  this  case,  we  have 

/.  A  =  {D\  +  a2  D,)  =  D,  [D]  +  a^) 

^-6  ((s(s2+a^))) 

=a:^  +  —  sin.  at  -\-~,  (1 —  cos.  a  t) 


2:;    .         ,,2  2;;'. 

—  sin.  a^-j 

a  a^ 

19* 


=XQ-f-  —  Sin.  at-\ 2^  sin.2  ^  at 


222  INTEGRAL    CALCULUS.  [b.  V.  CH.  IX. 

Linear  differential  equations  with  constant  coefficients. 

«/c'  b  sin.  m  r.  e^  ^^~'^^ «-  s  6  sin.  m  t — m  h  (cos.  m  f+c"') 

b  mb      / COS.  m  t      cos.  at\ 

ma-       m^ — a-  \     ni^  a^      / * 

Corollary.    When  m  is  equal  to  a,  the  value  of  JiT  becomes 

,,    6  ,  ^  btsm.at  26  ....    -        .  ,      . 

X=:  -;(l-cos.  at) — -— ^r-  =  -^sin.*a^(sin.ia  t-hat  cos. Aan. 
a^^  '      2a2      a^  ^  "^  ^     ' 

5.   To  integrate  the  differential  equation 

2>2  3._(^_j_5)  j)^  3._|_^  6  X  =  A  ^2_^A:  e'^'+Z  sin.  w  ^. 


a  —  b  a^  b^ 

2  A  (a+6)^        h  t^  2Ae^'  2Ae*' 


«2  62         '     a6   '   a'^{a—b)      b^  (a—b) 


(m — a)  (m — 6)      (m-a)  (a-b)      (m-b)  (a-b) 

l(ab  —  w^)  sin.  nt-[-nl  (a-\-b)  cos.  nt 
"^  (a2_^w-2)(62  +  w2) 

"^  (a2_|.yj2)(a_6J  "~  (62-(-;i2)  (^_5)  • 

When  ?»=:«,  the  terms  multiplied  by  k  become 
k  t  e'''        k  (e°'  — e^^)  ^ 
■^Z:^"  (a— 5)2     ' 

when  m'=.by  the  terms  multiplied  by  k  become 


a — b  (a — 6) 


2 


<^   196.]  LINEAR   DIFFERENTIAL  EC^UATIONS.  223 

Linear  differential  equations  with  constant  coefficients. 

6.  To  integrate  the  differential  equation 

D]  X  —  2  a  D,  X  -{-  a2  X  =  h  t^  -\-  k  e'"'  -{-  I  sin.  n  t. 

Ans.     X  =  Xq  e"^ -\- {xq — axQ)te''' 

+  "4  [6  +  4ai+a2  t^  +  2  (a t  —  S)  e'^'] 
k 

+  7-:; — — 7,r(«2-n2)sin.nM«2+w2)w^c«*+2aw(cos.w^-c«01. 

When  m-=.ay  the  terms  multiplied  by  A:  become 

i  kt^  e««. 

7.  To  integrate  the  differential  equation 

D"^,  X  -\-  a^  X  =:li  t^  +  k e"^'  +  I  sin.  n  t. 

X  h 

Ans.   x-=.  ~  sin.  a  t-\-XQ  cos.  a  t-\-  —  («2  ^2  —  4  siii.2  J  «  ^) 

A:(ac'"'-a  cos.  a^-m  sin.  «^)        l{n^m.at  —  as,\n.nt) 

When  w  =  a,  the  term  multiplied  by  I  becomes 

■— — -  (sm.  at  —  at  cos.  a  t). 

^Z  a^  ' 

8.  To  integrate  the  differential  equation 

D\  a:  =  x.  (95  a) 

Ans,     xz=l{x,^xi)  (e'+e-* )  +^  (^;+x;")(6'-e-0 
+  \  (^0—  ^0)  COS.  #  +  i  (2;;  — x;")  sin.  t 
=  i  (^0+  ^0 )  Cos.  ^  +  J  (2:0  +  aj'o')  Sin.  t 
+  J  (a^o  —  ^0)  COS.  #  +  J  (a:^  —  x'^')  sin.  t.     (96  a) 


224  INTEGRAL    CALCULUS.  [b.  V.  CH.  X. 

Linear  differential  equations  with  constant  coefficients. 

9.  To  integrate  the  differential  equation 

n\x  +  x=iO.  (97  a) 

Ans.     x  =  uxo-\-  Dl  u.Xo  +  B\  u,x"^^BtU.  x'^',     (98  a) 
in  which  w  =  Cos.  (/\/ J.  ^)  cos.  (\/ J.^).     (99  a) 

10.  To  integrate  the  differential  equation 

Dlx  =  x,  (lb) 

Ans.     x  =  uxo-\-  D^-'  w  .  2-0  +  D^^  u  ,  x^'  +  (Si-c,  (2  b) 
in  which,  when  n  is  an  odd  number, 

w  =  -lc*  +  2.^.c  ^       COS.  (^  sin. I    I,  (3  b) 

where  ^  denotes  the  sum  of  all  the  terms  which  are  obtained 
by  substituting  for  m  all  the  integers  from  1  to  J  (w  —  1)  in- 
clusive. But  when  n  is  an  even  number,  which  is  not  divisible 
by  4, 

(4  b) 


u 


=-|   Cos.  i-j-2  j^  .Cos. I  f  COS. l.cos, l^sm. I    I, 


where  2  denotes  the  sum  of  all  the  terms  which  are  obtained 
by  substituting  for  m  all  the  integers  from  1  to  J  (Jw — 1)  in- 
clusive.    When  n  is  divisible  by  4, 

(5  b) 

2r^                    rt^      /         2m^\          /     .    2m7t\-i 
w=-   Cos. ^+cos.^+2.2^.Cos.(<cos. j.cos.f^sm. )    , 

where  2  denotes  the  sum  of  all  the  terms  which  are  obtained 
by  substituting  for  m  all  the  integers  from  1  to  ^  n — 1  inclu- 
sive. 


§  196.]  LINEAR   DIFFERENTIAL    EQUATIONS.  225 

Linear  differential  equations  with  constant  coefficients. 

11.   To  integrate  the  differential  equation 

Dlx-\-x  =  0.  (G  b) 

Ans.     x=zuxo  +  D1-' u.x',  +  D^-'^u  . <  +  &c.,     (7 b) 

in  which,  when  n  is  an  odd  number, 

(8  b) 

where  2  is  used  as  in  (3  b).     When   n   is   an  even  number, 

which  is  not  divisible  by  4, 

(9  b) 

w=-rcos.  ^-|-2^.Cos.  Ucos.  (2»i-l)-)cos. /^sin.  (2»i-l)~)J 

where  2 .  is  used  as  in  (4  b).     When  n  is  divisible  by  4, 

M=-r^  .Cos.  (t  cos.(2  m-\ y^  COS.  (t  sin.  (2  ?»— 1)~  )  1(10^) 

where  2 .  denotes  the  sum  of  all  the  terms  which  are  obtained 
by  substituting  for  m  all  the  integers  from  1  to  ^  n  inclusive. 


(lib) 


12.    To  integrate  the  differential  equations 

D\x  +  ax  +  hy  =  X 

Dly  +  a'x+h'y=.  F, 

in  which  X  and  Y  are  functions  of  t. 

Ans.    In  this  case  we  have 

rz=.{D\.+  a){Dl  +  h')-a'h  (12  b) 

S  =  (s2  -f  a)  (52  +  h')  —a'b  (13  b) 

1=  [{D\+h')x'o+{Dl-{-^'D,)x,-byi-hy,D,]  ©     (14  b) 

ri=^[{D'l+a)y',-\-{D]-^aD,)y,—a'x',^a'x,D,]  0;  (lob) 


226 


INTEGRAL  CALCULUS.       [b.  V.  CH.  X. 


Linear  difterential  equations  with  constant  coefficients. 


and  if 


m  =  5  («  +  b'),      n^  =  i  («  —  h'Y  +  a'  b, 


{b'-jn-\-n)x'o-b  yl,  .  (h'-m-n)x!r-by^  .    ^   ,         . 


{b'-m-\-n 


V  = 


+ 


^o-b  !/o 


COS.  t\/{m-n)— 


(b'-7n-n)xo-b  i/o 


a — m4-7i)  y'o — a'  x'      .  , , 

^^'  -^^  0  sin.  t  \^{m—n) 


COS.  t/^{m+n) 
(16b) 


2  71  s/  (m — n) 

a — 771 — n)  yo  —  a!  x'o 


2  W  /y/  (77l-{-7l) 

a — 7n-\-7i)  y Q  —  a'  a:^ 


2  71 

a — 7n — w)  y^  —  a'  x^ 


2» 


sin.  t  s/  {ni-\-7i) 
COS.  t  s/  (m — 7%) 
COS.  t  \/  {m-\-n).  (17  b) 


If,  also,  Xj  it  are  the  values  of  X,  Y  when  ^  is  changed 
to  Tj  and  if  the  integrals  in  the  following  expression  are  taken 
relatively  to  t, 

.,       (6' 771 7Z)   X  &Y      .         ,  X       -,        .        X       /,o,v 

-■^■-       2„V (.«+»)         ^■°- ('-V('"+")   (18 b) 
-•^■-     3,.  V  (»>+»)      ^'°•('-^)^^ ("»+»)'    (19") 

we  have  a;  =  ^  -j-  ^',        y  = »;  -j-  j?'.  (20  b) 


«5>   197.]  LINEAR    DIFFERENTIAL    EQUATIONS.  2*27 

Linear  difFerenlial  equations  with  constant  coefficients. 


TIT,        /  V  •  sin.\/(m — 7i)  t       .  .,         . 

When  im — n)  is  negative, —r^ —  and  cqs.aJ {m-n]t 

s/  \m — n)  ' 

.       ,           .       S'\n.\/(?i — m)t      ,  ^  ,,        ,      , 

are  to  be  changed  to j—^ r^  and  Cos.  Aj{n-m)t.  (21  b) 

When 7w+nis  negative, tt—t- — -   ^nd  cos.  a/ (m +/i)  ^ 

s/ym-f-n) 

are  to  be  changed  to  — '     ,     , -■  and  Cos.\/-(m+w)^.  (^^  b) 

V — \m-\-n)  \      1     /      \        / 

1XTU  J  1    si"-  V  (^i — n)  t       .  ,/  X 

When  771  and  n  are  equal, '—  and  cos.vfm— 70< 

V  (?« — w) 

are  to  be  changed  to  t  and  unity.  (23  b) 

rxTu          I      •             sin.  \/(m-|-7^)f       ,  „      ,     x 

When  77^^-7^  is  zero, — ; — — -  and  cos.v  (tw+ti)^  are  to 

be  changed  to  t  and  unity.  (24  b) 

The  changes,  which  correspond  to  the  case  when  n  is  zero, 
are  easily  made. 

197.  Definition.  A  fluctuating  function ^\^  one,  which. 
constantly  changes  its  value  by  a  finite  quantity  for  an 
infinitely  small  change  in  the  variable,  alternately  in- 
creasing and  decreasing  without  ever  being  infinite. 

This  singular  function  is  of  great  use  in  the  integra- 
tion of  equations  which  involve  several  independent 
variables ;  there  is  no  name  in  general  use,  but  the  one 
here  adopted  was  given  by  Hamilton,  and  is  highly  ap- 
propriate. 

The  expression  sin.  a  x,  is  an  instance  of  such  a  function, 
when  a  is  infinite  ;  and,  in  this  instance,  it  is  noticeable  that 
the  mean  value  of  the  function  is  zero. 


228  INTEGRAL    CALCULUS.  [b.  V.   CH.  X. 

Linear  differential  equations  with  constant  coefficients. 

198.  Theorem.  Iff  denotes  a  function  of  a,  which 
is  continuous  and  finite  within  the  limits  a  and  b,  and 
if  N  is  a  fluctuating  function  of  which  the  mean 
value  corresponding  to  each  fluctuation  is  zero^  and  if 
the  integrations  are  performed  relatively  to  «,  we  have 

fi-N^f^^O.  (25  b) 

Proof.  Let  the  interval  between  the  limits  of  the  integra- 
tion be  divided  into  portions,  each  of  which  is  the  infinitely 
small  extent  necessary  for  a  single  fluctuation  ;  and  let  the 
limits  of  any  portion  be  ?  and  ?  -[~  i'  For  this  portion  we  may 
put 

«  =  i^  +  S  (26  b) 

and  the  corresponding  integral,  taken  relatively  to  £,  is 

But  by  (533)  of  vol.  1, 

M,  =U  +  i,/...„  D'f^,  (28 b) 

and,  by  definition, 

/5iV^  +  ,  =  0.  (29  b) 

Hence  (27  b)  becomes 

=  -^^  A  ■  N^+.  '"•  (30  b) 

But,  by  integrating  by  parts,  we  find 

f>.N,^^,j:^i3±^fA^ij^ilm.,  (31b) 

and  the  second  member  of  (31  b)  is,  evidently,  an  infinitesimal 


<§>  200.]  LINEAR  DIFFERENTIAL  EQUATIONS.  229 

^  Fluctuating  functions. 


of  the  (w-f  l)st  order,  and  (27  b)  is,  therefore,  an  infinitesimal 
of  the  same  order.  The  number  of  all  the  portions  of  (25  b) 
is  equal  to  ^  ,  and  therefore  the  sum  of  all  the  portions  (27  b) 

i 

is  an  infinitesimal  of  the  wth  order ;  that  is,  this  sum  is  infinitely 
small,  and  may  be  neglected,  which  gives  at  once  the  equa- 
tion (25  b). 

199.  Corollary.   If  we  take 

/■,=  /«  (32b) 

and  if  f  is  continuous  and  finite  throucrhout  its  whole  extent, 
(25  b)  gives 

/«  •  ^a  /c  =  0-  (33  b) 

200.  Theorem,    If  the  notation  of  %  198  is  adopted, 
and  if  x  is  included  between  a  and  6,  loe  shall  have 


/ 


l>N^,-^fa       =    f^f-      ILl 


.     .  (34  b) 


Proof.     In  the  identical  equation 

in  which  i  is  an  infinitesimal,  the  first  and  third  terms  of  the 
second  member  vanish  by  §  198,  when  this  equation  is  substi- 
tuted in  the  first  member  of  (34  b).     Hence  if 

a  =  «_2,  (36b) 

we  have 

/.  h  J^u-zfa^  ra+i    iV,-z  fu  ^  f-\-i  N,f+. 
J  a      a  —  X       J  « — i        « —  X        *I  — i         £       '    ^         ' 

in  the  third  member  of  which,  the   integrations  are  performed 
20 


230  INTEGRAL    CALCULUS.  [b.   V.  CH.  X. 

Fluctuating  functions. 

relatively  to  «.     But  f^-^^  differs  infinitely  little  from  f^,  and, 
therefore,  (37  b)  gives 

r\  I^=il±  ^  ff+i.  E^  .  (38  b) 

In  the  same  way,  when/^  is  unity,  and  x  is  zero, 

'  r    ^  =  r+\  ^  .  (39b) 

J  a  a.  J    — I      « 

which,  substituted  in  (38  b),  gives  (34  b). 
201.    Corollary.     Since  we  have 

/,QO  (a-2:)v-l p—  cc(a-a;)v-l 

f'  g>((x-a)v-l  — 

-^^  ~"  («_x)V  — I 

2  sin.  CO  (« — x) 


a  — X 


(40  b) 


the  first  member  of  (40  b)  may  be  substituted  for    —^    in 
(34  b),  which  gives 

^    ^      /»'      sin.  00  a       / . ,  ,  \ 
Si  ■  f\  eA(-«)v-i  /„  =  2  /.  J^     —^ .(41b) 

202.    Problem.    To  find  the  value  of 

/J!!!^.  (42  b) 

Solution.    If  we  put 

/5«z=«,  (43  b) 

we  have 

/•'    sin.  /5  a  /•'    sin.  a p'   sin.  a  ^  ...  ,. 

a  «  J  a         a  t/aa' 

that  is,  the  first  member  of  (44  b)  is  independent  of  the  value 


<5>  202.]  lilNEAR    DIFFERENTIAL    EQ,UATIONS.  231 

Fluctuating  functions. 


of  1^,  as  long  as  i^  is  positive  ;  so  that  if  yl  is  the  required  value 
of  (42  b),  we  have 

D^A=zO.  (45  b) 

We  have  also 

/•'   sin.  /?  a     _     p'    (1  +a^)  sin.  (J  g 
a  a  J  a  «  (I  -|-  o.^) 

/•'     sin.  ^  a         /•'    «sin.  ^a        ,.^,  . 
Hence,  by  putting 

£=/'^i;i-^,  (47b) 

''"''""         D,B=f'    ""'■/:  (48  b) 

P  J  a  l-|-a2  ^  ' 

/*i   «  sin.  /?  a 


whence,  by  (46  b), 

A  =  B  —  DIB.  (50  b) 

This  equation  may  be  regarded  as  a  linear  differential  equa- 
tion in  which  ^  is  the  independent  variable,  and  its  integral  is 

B=A  +  A'e^+  A"  e-'^  (51  b) 

in  which  A'  and  A"  are  arbitrary  constants.  The  values  of 
these  arbitrary  constants  may  be  determined  from  the  extreme 
values  of  Ds  B.  When  (^  is  infinite,  the  value  of  (48  b)  van- 
ishes by  (25  b)  ;  but  (51  b)  gives 

D^  B  =  A'  e^—A"  e-^ ,  (52  b) 

which  will  not  vanish,  when  ,^  is  infinite,  unless 

A'  —  0;  (53  b) 

whence  Ds  B  =  —  A"  e'? .  (54  b) 


23'2  INTEGRAL    CALCULUS.  [b.  V.  CII.  X. 

Fluctuating  functions. 

Again  when  ^  is  zero  the  value  of  (48  b)  is  ^  ;  and  although 
(i  may  never  be  supposed  quite  so  small  as  zero,  yet  when  it 
is  an  infinitesimal,  (48  b)  must  differ  infinitely  little  fiom  tt^ 
and  therefore,  by  (54  b), 

71  =  —  A",  (55  b) 

whence  D^  B  =z  rv  c~^,  (56  b) 

Finally,  the  comparison  of  (47  b  and  48  b)  gives,  by  {^Q  b), 
B^fiD^B  =  ^{\-c-l').  (57  b) 

Hence,  by  (51  b), 

A^.-=  f    ^l^    =     r'   ?i^^i^.        (58b) 

203.  Corollary.  The  substitution  of  (58  b)  in  (41  b)  gives 

f.  =  i-„f^-f'>.  e^^-'^'-V".  (59  b) 

provided  the  integral  between  the  limits  a  and  h  is  performed 
relatively  to  «. 

204.  Corollary.  \'i  fa  is  the  same  as  in  (33  b),  (59  b)  gives 

f.:^iKSLS\  ■  e'-^-"''-V<^  -=hfa',,  ^«^-"'^-'/»-  (CO  b) 

205.  Corollary.    In  the  same  way,  we  should  have 
L.f=  i-LA-  e«^-'')^-^  A .  ^ ,  (61  b) 
f..v  =  ~2n  f?  r,  •  e""-'-*^'-'  /.. ,,  ■'  (62  b) 

whence,  by  substitution, 


<§>  208.]  LINEAR    DIFFERENTIAL    EQUATIONS.  233 


Differential  equations  with  constant  coefficients. 

206.  Corollary.    In  the  same  way, 

={h')%:?  .y.i.,.,  eP(-«)+."(i'-«+.(-.)]v-i/^ .  ^ .  .^.(64  b) 

207.  Corollary.  The  successive  differentiation  of  (60  b) 
gives 

mfl  =  i-nJL  ^    (  -  1 )  1   /  e^(-°)^-l  /„  ;  (65  b) 

and,  in  the  same  way,  by  the  successive  differentiation  of 
(64  b),  the  factor  ;.  s/-\  is  introduced  under  the  signs  of  inte- 
gration for  each  differentiation  relatively  to  x,  the  factor  u\/-l 
for  each  differentiation  relatively  to  y,  &c. 

208.  Problem,  To  find  several  functions  JCt ,  Yi , 
(Sfc.  of  the  independent  variables  t,  a:,  y,  S^c.  which  sat- 
isfy given  linear  differential  equations  with  constant 
coefficients  between  various  differential  coefficients  cor^ 
responding  to  the  different  independent  variables^  and 
lohich  become  given  functions  X^ ,  Yq  ,  Sfc.  of  the  va- 
riable  X,  y,  <^*c.,  when  t  becomes  zero. 

Solution.  Let  Xt ,  IT^  &c.,  Xo ,  ITo ,  &c.  represent  the 
values  of  JT^,  F^ ,  &c.,  JTo,  Fq,  &c.  when  x,  y,  &c.  are 
changed  into  «,  ^,  &c.  ;  so  that  by  (64  b)  if  n  denotes  the 
number  of  the  variables  x,  ?/,  &,c., 

^'=(i^)V« .'ft  &c.  A .,„,  &c.  e['(-«)+"(y-«+&<=-]y-l  X,  (G6  b) 
&c. 

If  now  R  =  L  (67  b) 

represents  one  of  the  given  equations,  in  which  72  is  a  linear 
function  of  the  differential  coefficients  with   constant  multipli- 
20* 


234  INTEGRAL    CALCULUS.  [b.  V.  CH.  X. 

Differential  equations  with  constant  coefficients. 

ers,  and  Z.  is  a  given  function  of  t,  x,  y,  &c. ;  and  if  3L  de- 
notes the  value  of  L  when  x,  y,  &,c.  are  changed  to  «,  i^,  &c.  ; 
and  Iv  tlie  value  of  R  when  Xi  ^  Y^,  &c.  are  changed  to 
Xm  ITr ,  ^-c.,  and  Z)^  ,  Z>j^,  &c.  are  changed  to  ^\/-l,  ^</y^-l, 
&c. ;  the  equation  (G7  b)  is  changed  by  the  substitution  of 
(06  b)  into 

(68  b) 

(i^-7.)"/«.>>c...„&c.  .[^-(-«)+"(i/-^;+&-]v-i  (3tl-2L)=0, 

which  is  satisfied  by  putting 

^^IL,  (69  b) 

and  this  equation  involves  no  other  differential  coefficients  than 
those  taken  relatively  to  t. 

By  this  substitution,  therefore,  all  the  given  equations  are 
similarly  transformed,  and  the  problem  is  reduced  to  the  inte- 
gration of  several  linear  differential  equations  with  constant 
coefficients,  in  which  there  is  only  one  independent  variable; 
and  this  integration  is  performed  by  the  method  of  §  1 79  to 
"^  195,  The  functions  to  be  determined  are,  in  this  new  form, 
Xt ,  "^t )  &c.,  of  which  the  initial  values  are  ^o  >  ^o  j  ^c. 

209.  Corollary.  It  may  be  observed  that  for  a  complete 
solution,  the  initial  values  JTq  ,  Yq  &lc.  of  some  of  the  dif- 
ferential coefficients  D^  X^ ,  D^  Fj ,  &,c.  should  also  be  given 
functions  of  x,  y,  &c. 

210.    Examples. 

1.    Integrate  the  equation 

Dl  X,  +  D]  X,  =  0.  (70  b) 

Solution.    In  this  case,  the  substitution  {QQ  b)  gives 


<§)  210.]  LINEAR    DIFFERENTIAL  EQUATIONS.  235 

Dillerentiiil  equations  with  constant  coefficients. 

whence 

+  fr^  [e^^^-^-^  (l+V-3)e-Kv-l-3)^-^ 
*  -  i  (i-V-3)  ei(-^-i+3) ,  t\X';^  .      . 

If  the  values  of  Xq  j  X  o  and  X  o  are  written  as  follows, 

Xo=/,,     X  ;  =  />«/«,     X'o  =  Dlf'S;         (71b) 
we  have,  by  (60  b  and  Q5  b), 

/i/I^^o  e'(--«v-i  e*-(^-l±3).u  =/;;+j(^^_3),  ; 

whence  we  have 

^t  —  ^  {fz-t  +/.  x+i(l-V-3j£  +/•  x-|:i(l+v/-3)t) 

-i(/:-^  -  i  (  1- V-3)/.  ;+.kl-V-3).  -  i(  1+^-3 )/;+i  (1+^-3)0 

+^(/:'-^i(i+v-3)/.;'+i(i-v-3).-Hi-v-3)/.',+i(i+v-3)e). 

(72  b) 
2..  Integrate  the  equation 

a  b  Dl  X,  +  («  +  6)  i>; .,  X,  +  D]X,^  0.         (73  b) 


236  INTEGRAL    CALCULUS.  [b.  V.   CH.  X. 

Differential  equations  with  constant  coefficients. 


Ajis.    With  the  notation  of  (71  b), 

JC,=   —laf.-yt    -    6/x-at    +f'.-U    -f'.-at).   (74b) 

a — 6 

3.    Integrate  the  equation 

a'DlX,  -f  2aDl  ,X,  +  D',X,=:0.         (75b) 


t 


Ans.   With  the  notation  of  (71  b), 

X,  =/_.  +  ^/U.  +  at  D,./,_a..         (76b) 

4.    Integrate  the  equation 

aD,X,+  D,  X,  =  6"^^+"^  (77  b) 

Solution.    The  value  of  X^  in  this  case  is 

pmt-{-nC(  p — a tX-J  —  l  +  7i« 

m  -\-  a  X  /s/  —  1 
whence,  by  the  notation  of  (71  b), 

The  value  of  the  definite  integral   in  (79  b)  is  found  from 
the  equation 

.1-  y;2^  e;i(x-«)v-i  ew«  —  g^a;  ^  (80  b) 

which,  multiplied  by  -^  e™'^  and  integrated  relatively  to  x  gives 

i         r'^     ' ^-i_  —  i _ ;  (81b) 

^n  J  a.X      a  m'-\-a  i  s/ —  1  «  m'-\-a  n 

and  this  equation  divided  by  e'"'"'  is,  by  substituting  m  for  a  m', 


<§>  210.]  LINEAR    DIFFERENTIAL    EQUATIONS.  237 


Differential  equations  with  constant  coefficients. 


The  results,  successively  obtained  from  (82  b)  by  multiply- 
ing by  e'"'  ,  and  again  by  substituting  x — a  t  for  x,  reduce  the 
value  of  the  definite  integral  of  (79  b)  to 

f,mt-\-nx  pn{z — at) 

J- ,  (83  b) 

m  -\-  an 

and  the  value  of  X^  is  obtained  by  substituting  (83  b)  for  the 
definite  integral  ;  so  that 

X,z=/_,  +(83b).  (84  b) 

Corollary.     When         m  =  —  a  7i,  (85  b) 

(83  b)  is  reduced  to  t  c'^(,^-«".  (86  b) 

5.  Integrate  the  equation 

aD,X,  +  D,X,=:z  tx.  (87b) 

Ans.  X,-=if,^a  «+^— 2  —  J  a;2  ^  +  J  « .T  #2  _  ^  «o  ^3.     (88  b) 

/4  a 

6.  Integrate  the  equation 

Dl  X,  +  D\  X,  =  (x2  +  ^2)  ,xt,  (S9  b) 

(90  b) 

7.  Integrate  the  equation 

aD.vX^-^b  D,X,-\-D,X^=lc'^^'+^y-^^t,  (91  b) 

7  ph  .V-\-k  y  r,,7n  t „—(h  a+k  b)n 

ah  -f-o  k  -j-  m  ' 

which,  when  m  =  —  ah-\-hk  (93  b) 

is  reduced  to 

X,=/,._,,,  y_j,+/^e''-^  +  Ay-(A«  +  i6;«.  (94b) 


238  INTEGRAL    CALCULUS.  [b.   V.  CH.   X. 

Diirerential  equations  with  constant  coefficients. 


211.  The  integration  of  linear  differential  equations, 
in  which  the  coeflicients  are  not  constant,  can  only  be 
performed  in  some  particular  cases,  some  of  which  will 
be  found  in  some  of  the  following  chapters. 


«5>212.]    DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER.         239 
Equations  of  the  first  order. 


CHAPTER   XI. 

INTEGRATION     OF    DIFFERENTIAL    EQ,UATIONS    OF    THE 
FIRST    ORDER. 

212.    To   integrate  a  given  differential  equation  of 
the  first  order,  between  two  variables  x  and  t. 

Solution.    Let  t  be  the   independent  variable,  and  let  the 
value  of  Z>jX  be  found  from  the  given  equation  in  the  form 

M 

D,x=--.  {95  b) 

The  integral  of  this  equation   must  involve  an  arbitrary  con- 
stant a,  from  which  the  value  of  a  can  be  found   in  terms  of 

t  and  X  in  the  form 

a=A,,  (96  b) 

in   which  A^  is  a   function   of  t  and  x.      The  differential  of 
(96  b)  gives 

^-DxA,.D,x  +  D,A,,  (97  b) 


DA 

JJ^XZZZ  

Hence,  by  (95  b), 


^'^=-i^-  (^^'') 


D^_ll_±M.  (90M 

in  which  x  is  wholly  arbitrary,  and  may,  therefore,  be  taken  of 

such  a  value  that 

^N-DxA,,  (Ic) 

which  gives  ^  M—  D^  A^;  (2c) 


240  INTEGRAL    CALCULUS.  [b.  V.   CH.   XI. 

Equations  of  the  first  order. 


whence,  by  the  elimination  of -4,, 

Dl,A,^D.(>-N)^D,(xM).  (3  c) 

There  is  no  general  process  of  finding  a  value  of  ^  which 

will  satisfy  (3  c),  and  this  problem  must  be  solved  in  each  case 

by  the  exercise  of  the   ingenuity.      When  the  value  of  ^  is 

found,  (1  c  and  2  c)  give 

a^A,=f,{^M)^f,  [7.N),  (4c) 

in  which  a  is  the  arbitrary  constant. 

213.  Corollary.  An  arbitrary  function  of  x  will  be  added  to 
the  third  member  of  (4  c)  to  complete  the  integral,  and  an 
arbitrary  function  of  t  to  the  fourth  member  of  (4  c).  But 
these  arbitrary  functions  are  at  once  determined  by  the  con- 
ditions that  the  third  and  fourth  members  are  equal. 

214.  Corollary.  The  value  of  a  is  usually  determined  by 
the  condition  that  x  is  to  have  a  certain  value  x^ ,  when  t  be- 
comes T.  Ifj  then,  A^  denotes  the  value  of  At  when  t  and  x 
are  changed  to  ^  and  x^  ,  (4  c)  gives 

A,-A^  =  0.  (5c) 

215.  Corollary.  It  is  often  the  case  that  the  given  equation 
is  such  that  it  cannot  be  reduced  to  the  form  (95  b),  and  in 
this  case  the  whole  process  must  be  Jeft  to  the  skill  of  the 
geometer. 

216.  Corollary.   If  iW  and  N  are  such  functions  of  x  and  t, 

that 

M=3Ia:3l,       iV=iY^-iV,  (6  c) 

in  which  31x  and  Nx  are  functions  of  x  alone,   and  M^  and 
Nt  are  functions  of  t  alone,  the  value  of  ^  may  be  assumed 


<5»217.]  LINEAR    DIFFERENTIAL    Eq,UAT10NS.  241 

Homogeneous  equation. 

For  this  assumption  reduces  the  two  last  members  of  (3  c)  to 
zero.     The  equations  (4  c  and  5  c)  give 

217.    Corollary.  When  M  and  TV  are  homogenous  functions 
of  the  same  degree  m,  the  vahie  of  ^  is 

^=i{Nx  +  Mt)-'.  (9  c) 

Hence      7r^  =  Nx-{-  31 1  (10  c) 

2>.r  ^  =  —  A2  (iv+  X  D.vN+t  Dv  M)  (U  c) 

A  ^-  =  —  ^-^  {M  -\-x  D,  N  +  t  D,  31)  (12  c) 

Dx  {^-  31)  =  —  -^^  (31  N -\-  31 X  Ds  N'-N X D,v31 )    (13c) 

D,(^-N)  —  —  7.'i(3IN—3ItD^N+NtD,31).  (14c) 

_,        ,  .  X 

But,  by  puttmg  y  =  — 

31,  r  ■  r  . 

the  expression  —   becomes  a  function  of  i/  alone,  which   may 
be  denoted  by  31',  whence 

D.r  M'=D, M.  D,y=  \  D, M'=  ^         (15c) 

X 


D,  31'  =  I),^  31' .  D,  ij  =z  —  —  D^  31' 
D,  31         m  31 


(16c) 


and,  therefore, 

X  D:cM=  —  tD,31+m3I;  (17  c) 

and,  in  the  same  way, 

X  D:,Nz=z  —  tD,N+mN,  (18c) 

21 


242  INTEGRAL    CALCULUS.  [b.   V.   CH.   XI. 

Infinite  number  of  multipliers. 

which,  substituted  in  (13  c),  give  by  (14  c) 
Dx{}  M)=—).\MN-Mt D,N+Nt D,M)=DlxN).  (19c) 
'  Hence  (3  c)  is  satisfied,  and  (4  c)  gives 

_  p        M         _  ^        N 

""-J  tNx-YMt-J:cNx+Mt'  ^^^^^ 

218.  Corollary.    If  h  is  any  function  whatever  of  a,  and  if 
Bt  is  the  same  function  of  ^^ ,  (96  b)  gives 

b  =  B,.  (21  c) 

It  may  be  shown,  precisely   as  in  §212,   that  if  ,"  is  such 

that 

uN=^D^.Br,  (22  c) 

u  will  be  a  value  of  ^  capable  of  satisfying  the  equation  (3  c). 
If,  however,  b'  is  the  differential  coefficient  of  b  taken  rela- 
tively to  a,  and  if  ^^  is  the  same  function  of  A^  which  b'  is  of 

a,  we  have 

Da;B:=  B[D^A,,  (23  c) 

whence  (22  c  and  1  c)  give 

^cN—^-B'tN    or     u  —  7.Bl\  (24c) 

that  is,  the  product  of  any  value  of  ^  by  any  function  what- 
ever of  A^  is  itself  another  value  of  i. 

219.  Corollary.  Whenever  M  and  N  can  be  separated 
into  such  portions  M ,  31',  31'",  &.C.,  and  N',  N",  N'",  &c., 
that  the  equation  (95  b)  can  be  integrated  when  for  M 
and  N  are  substituted  31  and  iV',  or  M"  or  iV",  &c.,  the  inte- 
gral of  the  equation  itself  is  often  readily  obtained.  For  this 
purpose,  let ;.'  and  A[  represent  the  values  of/  and  A  ^  which  cor- 
respond to  M'  and  N',  '■"  and  A'\  those  which   correspond  to 


<§>  220.]    DIFFERENTIAL  EQ,UATIONS  OF  FIRST  ORDER.       243 


Equations  of  the  first  order. 


M"  and  N",  &c.  it  is  necessary  to  find  functions  ^',  (f"  &c. 
o^  A't,  A'l  &;c.,  which  will  satisfy  the  equation 

^■'  cp'.  {A',)  =  ;."  /.  (A;)  =z  x'"  y";  (^V')  =  &c.    (25  c) 

For  if  the  value  of  each  member  of  (25  c)  is  denoted  by  ij  we 

shall  have 

A  J/=:  /'  /.  {A\)  31'  -f  k"  ^p"  {A';)  M"  +  &c.      (26  c) 
^  iV=  a'  <f'.  {A\)  N'  +  ^"  9"  [A]')  N"  +  &c.    (27  c) 

But,  by  the  preceding  corollary, 

D,  [>.'  ^p'.  {A[)  N']  =  Dj;  [^.'  cp'  {A',)  J/],  &c.     .   (28  c) 
and  therefore  ;.  satisfies  (3  c). 

220.     Examples. 

1.   Integrate  the  equation 

{tX'+T)D,xJ^X+xT'  =  0,  (29  c) 

in  which  JC  is  a  given  function  of  x,  and  JC'  its  differential 
coefficient;  and  T  is  a  given  function  of  t,  and  T'  its  diffe- 
rential coefficient. 

Solution.     In  this  case, 

M  =  X+xT' 
N=  T+tX' 
D,N=  T'  +  X'  =  Z>,.  31, 
and,  therefore,  (3  c)  is  satisfied  by 

;.  =  1. 

Hence  the  required  integral  is 

—  Xt  +  xT;  (30  c) 


244  INTEGRAL    CALCULUS.  [b.  V.  CH.  XI. 

Equations  of  the  first  order. 

or  if  X   and  ^  are   the  values  of  X  and  T  when  t   and  x 
are  changed  to  ^  and  x^ , 

Kt-^x^K  —  Xt^xT.  (31c) 

2.  Integrate  the  equation 

{t  cos.  X  -|-sin.  €)  D,,x  -\-  sin.  x  -f-  x  cos.  ^  r=  0.       (32  c) 

Ans.    Tsin.  x^-j-.x   sin.  t  ^r:  i  sin.  x-j-x  sin.  ^.    (33  c) 

3.  Integrate  the  equation 

x«  £'  D^x-{-  k  x'^'  ^^'rz:  0.  (34  c) 

Corollary.     When        «  —  «'  -}-  1  =  0, 
the  answer  is 

when  J/  _  5  _f_  1  —  0, 

it  is  ^a-a'+l  _  ^a^a'+l  f 

____x__+,,og._  =  0;  (37  c) 

and  when  both  these  conditions  are  satisfied,  it  is 

X  T^  +  ^  x^,  =  0.  (38  c) 

4.  Integrate  the  equation 

t  D,x=:x  +  \/  (x2  +  f2).  (39  c) 

Solution.    This  is  a  homogeneous  equation,  and  (9  c)  gives 

^-'  =  M  t+N  X—tX—t  X+t  \/(x2+i2)  —  t  ^(:t2+^2)  . 

hence,  by  (20  c),  the  integral  is 

=  \og.W(x^  +  fi)-%l  (40  c) 


<§>  220.]    DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER.       245 


Equations  of  the  first  order. 


or  V(xa  +  ^2)__3.— ^(3,2_|_r2)_^^^  (4lc) 

or  t^zzz  a^-{-2ax,  (42  c) 

in  which  a  is  the  arbitrary  constant. 


5.    Integrate  the  equation 


6.  Integrate  the  equation 

D.x^j——   log.-.  (45  c) 

Ans.      {;-iY={;^Y.     t46c) 

7.  Integrate  the  equation 

(l+^log.  ^)A^=:^-(l  +  /^Iog.  j^         (47c) 

^"^-   (r)     =(f)     •    (''^'^) 

8.  Integrate  the  equation 

(A  r  x"+i  +  A'  r'  x"'+i) 
-|-(A: ^"»+i  a;'*  +  k'  r'+^  z")  Z>,  X  =r  0.  (49  c) 

Solution.    This  is  a  case  of  §  219,  and  by  putting 

M'  =hr  x''+\     M"  —  h' r'  x"'+i ,  ?     /        \ 

M'  =  k  «'«+!  X",     M"  z=  A;'  ^''+1  x"'  j  ^     ^      *^^ 

we  have 

'  '  (     f5l  c'i 

Al  -  log.  ^'^  x^      A'/  =  log.  <'''  x^',  s 

21* 


246  INTEGRAL    CALCULUS.  [b.  V.  CH.  XL 

Equations  of  the  first  order. 

and  if  «  and  a  are  taken  to  satisfy  the  equation 

which  gives 

ah  -\-  ?n  =  a'  h'  -\-  in',     «  ^  -f  w  —  «'  A;'  +  n'         (53  c) 

_   {m  —  m')  k'—  {n  —  n')  li' 


h<  k  —  h  Id 
I (m — m')  k  —  {n  —  n')  h 


(54  c) 
(55  c) 


h'  k  —  lik' 
we  may  put 

l-\  __  ^aA+m+l   2:«^+n+l    ^  (56  c) 

and  the  integral  of  (49  c)  becomes 

9.  Integrate  the  equation 

{^  ax  t  +  2h  e-)  D^x  +  ^  ax^  -\-^  h  xt  =  0. 

Ans.    a  x3  t^-\-h  fi  x^=a  x^  r'^+b  x|  t^. 

10.  Integrate  the  equation 

(3  a  x3  ^3_|_2  6 1)  D,  x+2  a  x^  t^+S  b  xznO. 

Ans.     a  (x3  i2_2;3  t2)-[_6  log.  -lA  —  o. 

11.  Integrate  the  equation 

(^hx  +  kt-\-a)  D,x  +  h'x-\^k'  t  +  a'  =z  0.      (58c) 

Solution.    Put,  in  this  equation, 

x  =  x'-f-/?,       t=:t'  +  »^  '      (59  c) 

and  we  have      DtXz=.  D^x'  =z  D^,  x',  (60  c) 

whence  (58  c)  gives  (61  c) 

(hx'+kt'+h(i+kcc-{.a)  D,.x'-{-h'x'-\-k't'+h'  ?+k'a+a'=0; 


<5>  220.]    DIFFERENTIAL  EQ,UATIONS  OF   FIRST  ORDER.       247 

Riccati's  equation. 

and  if  «  and  ^  are  taken  such  that 

h^-^ka+a  —  0,      h'(i  +  k'a+  a'=:0,         (62c) 
(61  c)  becomes 

(h  x'-^k  t)  D,,%'-\-  h'  X'  +  k'  t  —  0,  (63  c) 

which  may  be  integrated  like  any  other  homogeneous  equation. 

12.  Integrate  the  equation 

n,x-\-  Txz=z  T',  (64  c) 

in  which  T  and  T'  are  functions  of  t. 

Solution.     Let  i'  =  ft  T,  (65  c) 

whence  D,t'  —T  (66  c) 

D,x  =  D,.xD,  t'—  T  D,.x;  (67  c) 

T' 

and  if  T"  denotes  the  vakie  of  —   when   t'  is  substituted  for 

t,  (64  c)  gives 

D,.x-{-x=:z  T",  (68  c) 

which  may  be  integrated  by  the  processes  of  the  preceding 
chapter,  since  it  is  linear,  with  constant  coefficients.  The  inte- 
gral is,  if  t'  is  the  value  of  t'  when  t  becomes  t, 

x  =  x^  c-f '-"  +  e-''  /V-  2'"  t" 

'  =  x^  e-'f'^  ^+  e-^7;  T'  e-f  ^.  (69  c) 

13.  Integrate  the  equation 
kx  a{t+h'f 


D,x  + 


t-^h~'  (f-{-  hy 


^^^-  ^-^r^Tw+      {k'+i)(t+hr      • 

14.   Integrate  the  equation 

D,x-{-  hx^  =  k  r,  (70  c) 

which  is  called  Riccati's  equation. 


248  INTEGRAL    CALCULUS.  [b.  V.  CH.  XL 


Riccati's  equation. 


Solution.    Let  x'  and  t'  be  so  taken  that 

-  +  - 

ht     ~  x't 
and  we  have 

t' 


^=TT-+T77F'     t'  =  tm+^l  (71c) 


D,x=:Dr  X'.  D,  <'=(;«+3)r+2  D,,x'—{m+'^)-D,.x'  (72c) 

1  2  D,x' 

D,x- 


h  t^  X'  f  X'2  ^2 

——  — ^ -^  —  (?/i  +  3)  — -V      (73  c) 


1 


^^  ^'=  Ti,  +  17^3+::;^  (74  c) 


A—  m+3^  ^' A  a;'       Ax'2f^ 

Hence 

h  li  1  li  ?^+4 

D„x^'+h'  x'^=k't"^\  (79c) 

which  is  of  the  same  form  with  the  given  equation.  Hence  if 
Riccati's  equation  can  be  integrated  for  any  value  m'  of  m,  it 
can  also  be  integrated  for  the  value  m  determined  by  (78  c) ; 
and  if  it  can  be  integrated  for  the  value  m,  it  can  also  be  inte- 
grated for  the  value  m'. 

Let  i  be  determined,  so  that, 

4z 


m  z= 


2f-i-l   ' 


(80  c) 


<5)  220.]      DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER.       249 

Riccati's  equation. 

and  (78  c)  gives 

'^^---27T3=--2lktT)Tl^  (^^^) 

so  that  m'  is  obtained  from  m  by  increasing  i  by  unity.  Hence 
if  the  equation  can  be  integrated  for  any  value  of  /,  it  can 
also  be  integrated  for  the  values  of  i,  which  are  greater  or  less 
by  unity,  and  therefore  for  any  value  which  differs  from  i  by 
any  integer  whatever. 

But  when  2  =  0,  (82  c) 

we  have  m  =  0,  (83  c) 

and  Riccati's  equation  becomes 

Dtx  +  h%^  =  k,  (84  c) 

the  integral  of  which  is 

t-r-  r  ^  -  J-  W  Wf'+Wll)s/{h-h  X^  )^ 

so  that  Riccati's  equation  may  be  integrated  whenever  i  is  an 
integer  either  positive  or  negative. 

When  i  z=z  ±  x,  (86  c) 

we  have  w  =:  —  2,  (87  c) 

and  therefore  this  case  would  only  be  obtained  from  the  pre- 
ceding, by  an  infinite  succession  of  substitutions.  This  case, 
however,  admits  of  direct  integration,  for,  by  the  substitution 

^=1^  +  7-  (^^'^> 

Riccati's  equation  becomes  in  this  case 

^•2  2>^  X'  +  x'2  =  yt  <2,  (89  c) 

which  is  homogeneous,  and  its  integral  is 
[2a:+^-V(l+4A')][2z^+r+V(l+4X0]  _  /r  w(i4-4ft) 
[2x+/+^V(l+4  /c)][2a;^+T-r>/(l+4  k)]  "  \t  )        (90  c) 


250  INTEGRAL    CALCULUS.  [b.  V.   CH.  XI. 

Equations  of  the  first  order. 

15.  Integrate  the  equation 

P  =  0,  (91  c) 

in  which  P  is  a  given  function  of  D^  x. 

Solution.  By  solving  the  equation  (91  c)  relatively  to  D^  x, 
each  of  its  values  will  be  found  to  be  a  constant,  one  of  which 
we  may  denote  by  7n. 

Hence  DtX  =  m,  (92  c) 

Avhence  x — x^  =  7n  (t  —  t)  (93  c) 

and  X — X 

m  =  j--f=D,x,  (94  c) 

and  the  second  member  of  (94  c)  may  therefore  be  substituted 
for  Dc  X  in  (91  c)  ;  and  if  Q  represents  the  value  of  P  arising 
from  this  substitution,  the  integral  of  (91  c)  is 

Q  =  0.  (95  c) 

16.  Integrate  the  equation 

D^  x^  =  a9. 

Ans.    {x — x^Y=^a^{t — t)2. 

17.  Integrate  the  equation 

\^{\+D,x^)z=za+hD,x. 

Ans.     ^[{x-x^Y+it-^Y]=<t-~^)+K^-x^)- 

18.  Integrate  the  equation 

2>,  X"  z=  T,  (96  c) 

in  which  T  is  a  function  of  t. 

Ans.     x,^x^=  /;  V  T,     (97  c) 

or  the  equation  which  is  obtained  by  freeing  (97  c)  from  radi- 
cals. 


§  222.]    DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER.       251 

Equations  of  the  first  order. 


19.  Integrate  the  equation  ^ 

D^  x^  =  t. 

4.  3 

Ans.     {x—x^-\-r'')  =  U  <  • 

20.  Integrate  the  equation 

P  =  t,  (98  c) 

in  which  P  is  a  given  function  of  2)^  x. 

Solution.    By  putting    p  z=z  D^x^  (99  c) 

(262)  gives 

x—f,D,x—f,p—f,ipB,t  —  pt—f,  tD,p 
=  pt-f,tD,p=pt-f,.P,  (Id) 

and  the  integral  is  obtained  by  eliminating  p  between  (Id)  and 
the  equation  obtained  from  (98  c)  by  changing  DiX  io  p. 

21.  Integrate  the  equation 

D  X 

Ans.     It  is  the  equation  obtained  by  eliminating  p  and  ^ 
between  the  equations 

t  =i?4"  e^  -{-sin  p 

Tr=p^-f-c^T-f-sin.  p^ 

X X^^^p  t p    T J  (p2 p2^ gP_j_gP^_|_COS.^ — COS.p^. 

22.  Integrate  the  equation 

t-\-  a  D.xzzzb  s/  {\  +  D,  x^), 

Ans.  It  is  the  equation  obtained  by  eliminating  p  and  p^  be- 
tween the  equations 

t  +  ap  —  6\/(l+p2) 

r+ap^=h^{l+p^~) 

x-x^  =  pt-p^r-^ia{p^-p^^)-ip^/(l+p^) 


252  INTEGRAL    CALCULUS.  [b.  V.  CH.  XI. 

Homogeneous  equations  of  first  order. 

23.    Integrate  the  differential  equation  of  the  first  degree, 
which  is  homogeneous  in  reference  to  the  variables  x  and  t. 

Solution,    Let        y^z-y     DtX=p^  (2d) 

which  gives  x  =z  1/ 1  {^  ^) 

Dy%  =  y  Dyt  +  t^D,xDyt=pDyt  (4  d) 


Dyt     _  1 


t  p—y 


(5d) 


^'S-t=f    -^   ,  (6d) 

•7  y  p    y 

But  the  substitution  of  (2d)  in  the  given  equation  reduces 
to  an  equaticm  containing  only  p  and  y ;  hence  the  integral 
(6  d)  is  readily  obtained,  and  the  required  integral  is  obtaine'd 
by  eliminating  p  and  y  from  (3  d,  6  d)  and  the  given  equation 
in  the  form  to  which  it  is  reduced  by  the  substitution  of  (2  d). 

24.  Integrate  the  equation 

Ans.  The  equation  resulting  from  the  elimination  of  p  and 
p    between  the  equations 

X  z=z  p  t  ~\-n  t  ^/  {\  -j-  p^) 

ix//i+z!\  _  (Pr  +  ^i^+p?)  Y- 

25.  Integrate  the  equation 

xz=zPt+Q,  (7d) 

in  which  P  and  Q  are  functions  of  D.  x. 


<5>  220.]    DIFFERENTIAL  EQUATIONS   OF  FIRST  ORDER.      253 


Equations  of  the  first  order. 


Solution.    Let  p=  D^x,  (8  d) 

and  the  differential  of  (7d)  gives 

DpX=2D^x.Dpt=p  Dpt 

=  P  Dpt  +  tDpP  +  DpQ,  (9d) 

or  (p^P)  Dpt  —  tDpP^DpQ,  (10  d) 

which  is  a  linear  equation  of  the  first  order,  by  taking  ^^  as  the 
independent  variable.  The  integral  of  (10  d)  is  an  equation 
between  t  and  p  from  which  p  can  be  eliminated  by  means  of 
the  given  equation. 

26.    Integrate  the  equation 

^       *  711''         ' 

Ans.     The  integral  is  found  by  eliminating  p  and  p    be- 
tween the  equations 

x~{p—i)t-\-e'^^ 


,  ^  nip  Jn(p — pj) 

t—m{p—p^)e    ^  =Te    ^    ^^' . 


27.    Integrate  the  equation 

x  =  tD,x  +  P,  (lid) 

in  which  P  is  a  function  of  Z),  x. 


X — x_ 


Ans.      If  P    denotes  the  value  which  P  obtains    when 


"^  is  substituted  for  D^x^  the  required  integral  is 

tx^-xr={^t--r)P^,  (12d) 


22 


254  INTEGRAL    CALCULUS.  [b.  V.  CH.  XI. 

Equations  of  the  first  order. 

28.  Integrate  the  equation 

x=t  DtX  +  n\/(l  +  D,  x2). 

or  {tx^—x^Yz=in^{t—'tY-\-n^{x^x^Y. 

29.  Integrate  the  equation 

D,xz=i  (Af'+Bx')^"^.  (13d) 


a 


Solution.     Let  uz=zxt  a  , 

a 
or  X^=iUt~b  ; 

1  _  1    ffl^ 
z=i{A+B  uy    «  «  S 


and 

1 

h  D,u 

'    ""6(^  + 

■Bu'')l~a     . 

—  au 

log. 

t        ru 

b 

(A+Bu'y' 

1 
~« —  au 

in  which 

u^  =  x^r- 

a 

30. 

Integrate  the 

equation 

t  DtX  —  X 

-r(. 

/•T 

\/(x'+t^W{D,x^+l)      L\F.{x^+t^) 
in  which  /.  and  F.  are  any  given  functions. 


(14  d) 
(15  d) 


y+i]  *(i6d) 


<5»  220.]    DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER.       255 


Equations  of  the  first  order. 


Solution.   Let  r  and  (p  be  taken  so  that 

t  :=:  r  COS.  (p,      x  z=.  r  sin.  g),  (17  d) 

whence  ^2^^2  +  ^2,     tan.  <^  =  ?  (18  d) 


(19d) 
(20  d) 


Dt  x=z  sin.  cp  DtV  -{-  r  cos.  <jp  Z)^  <3p 
1  z=z  COS.  (p  Dt  r  —  r  sin.  (p  Dt  (p 

t  D^x  —  X  =z  r^  Dt  <f> 
D,x^  +  1  =  2>,  r2  +  r2  Z>,  92, 

and  (16  d)  becomes 

or  f.idin.  (p.Dt(p  =r-^  V^r  .F.r^,  (22  d) 

and  its  integral  is 

/;/.tan.,=/;^^:  (23d) 

in  which  2:  „         „    ,     „  /r...  iv 

tan.  9)^  =  -?,     r|=.a:|+T2.  (24  d) 

31.  Integrate  the  equation 

Ans.     By  the  notation  of  the  preceding  example, 

r,  rV[r2-(/.r^n- 

32.  Integrate  the  equation 

t  Dt  X  —  X  X 


256  INTEGRAL    CALCULUS.  [B.  V.  CH.  XI. 

Equations  of  the  first  order. 

Ans.     By  the  notation  of  example  30, 

r*cp  V  [1  —  (/-tan.  cp)2]  _  J        21 . 
J  (p^  f.  tan.  (f  ~~        *    r^ ' 

33.    Integrate  the  equation 

tPt^  —  ^  _r-/        M_Y_.-\-h 

^{t^—x^)s/{V—D,x^~)~L\F{t^-x^)}         A     -y    I 
Ans,     By  putting 


X 

r^  z=  t^ — x^.     Tan.  OP  = — 

X 

y.2  __  t2  —  372     Tan.  CO  ^r  -^ 

T  T  '1: 


j-   (26  d) 

J 
the  integral  is 

/;./.Tan..=/;.^^  (2rd) 

34.    Integrate  the  equation 

\^{x^-t2)s/{D,x^-l)      L\F.{x'^^t2)/       ^J    '^'^^    ^ 

Arts.   By  putting 

j,2  zz:  x^  -\-  t^ f       p  z=z  X  tf 

r|  =  a?|  +  '^^»      Pr^  ^t  '^  ' 
the  integral  is 

£rF.r^=fP  f.p.  (30  d) 

221.  Problem.  To  integrate  several  differential  equa- 
tions between  several  variables  and  their  differential  cO' 
efficients  taken  with  respect  to  one  of  them  regarded  as 
the  independent  variable. 


(29  d) 


<5>  222.]    DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER.       257 

Equations  of  the  first  order. 

Solution.  By  taking  the  successive  differentials  of  these 
equations  with  respect  to  the  independent  variable,  as  many 
new  equations  may  be  obtained  as  may  be  necessary  to  elim- 
inate from  them,  combined  with  the  given  equations,  all  the 
variables  but  two,  of  which  one  is  the  independent  variable, 
together  with  their  differential  coefficients.  The  resulting 
equation  will  be  an  equation  between  these  two  variables,  and 
the  successive  differential  coefficients  of  one  variable  taken 
with  respect  to  the  other,  which  is  the  independent  variable. 

In  most  cases,  however,  the  integration  can  only  be  obtained 
by  some  ingenious  device.  Examples  of  this  problem  will 
occur  under  the  subsequent  problem. 

222.  Problem.  To  find  a  function  v  of  several  inde- 
pendent variables  t,  x,  t/,  Sf'c,  which  satisfies  a  given 
differential  equation  of  the  first  order^  and  becomes  a 
given  function  of  the  variables  x^  y,  (J'c,  for  a  given 
value  T  of  the  variable  t. 

Solution.  If  D'  denotes  the  differential  coefficient  with 
reference  to  the  given  function  of  the  variables  a-,  ?/,  (S:c.,  and 
if  s,  p,  q,  &c.  denote  the  differential  coefficients  D^  U',  D  .  V, 
2>y  V,  &,c.,  we  have    ' 

D'  H'  =p  D'x-\-q  D'  1/  +  &i.c.  (31  d) 

If,  moreover,  x,  i/,  &c.  instead  of  being  independent  of  ^,  were 
assumed  to  be  certain  functions  of  t,  we  should  have 

D,^  =  s+p  D,z  +  q  D,ij  +  &,c.,  (32  d) 

the  differential  coefficient  of  which,  relatively  to  D'  is 

D  D,^=p  D'  D,x  +  q  D'  D,7j-\-  &LC. 

+  D's-\-D,x.D'p  + D,y.D'q  +  &LZ.    (33d) 
22* 


258  INTEGRAL    CALCULUS.  [b.  V.  CH.  XI. 

Equations  ofllie  first  order. 

But  the  differential  coefficient  of  (31  d)  relatively  to  t  is,  by 
this  assumption, 

+  D,p.D'x  +  D,q.D'i/-\-&Lc.;    (34 d) 
and  the  difference  between  (33  d  and  34  d)  is 

0  =  D'  s  +  D,  X  ,  D'  p  +  D,7/ .  D'  q  +  6zc. 

—  D,p.D'x-^D,q.D'y-{-&DC.       (35  d) 

If  the  given  differential  equation  becomes  by  the  substitution 
of  5,  p,  q,  &,c.  for  the  differential  coefficients  of  v^, 

22  =  0,  (36  d) 

its  differential  coefficient  is 

0=:Z>^  R  .  D'  ^+D,  R .  D's+Dp  R  .  D'p+D^  R  .  D'q+&.c. 
+/>^.  R  .  D'x^Dy  R  .  D'i/+&LC.,  (37  d) 

which  becomes,  by  the  substitution  of  (31  d), 

0=D,R.D's+Dj,R.D  p-{-D^R.D'q+&LC. 

+(p  D^  R+D^^R)D'x+{q  D^  R-\-D,R)D'y+&LC.  (38  d) 

But  (36  d)  is  the  only  given  equation  between  the  quantities 
t,  X,  y,  &/C.,  s,  p,  q,  &c.,  and  cannot,  therefore,  determine 
more  than  one  of  the  differential  coefficients  D' s,  &c.  in  terms 
of  the  others ;  so  that  the  value  of  this  differential  coefficient, 
determined  from  (38  d),  must  be  the  same  with  that  given  by 
(35  d) ;  and,  consequently,  the  product  of  (35  d)  by  D^  R 
must  coincide  with  (38  d).     Hence 

I  D,x  D,  y 


D,R~   D,R   -   D^R 


=  &c. 


jjD^R^D^R-  qD^R+D;R-^'''^^^^''^ 


^j   -•'     I     ^x   ■-■*'  \l    -^  yj 


<5>  223.]  LINEAR    DIFFERENTIAL    EQ,UATIONS.  259 

Equations  of  the  first  order. 

and  (32  d)  gives,  by  the  theory  of  proportions,  each  of  these 
fractions,  equal  to 

s  D,  R-\-p  Up  R-\-q  A,  Ii-{-&^c.   '  ^        ' 

The  equations  (36  d,  39  d,  and  40  d)  may  then  be  regarded 
as  several  equations  of  the  first  order,  with  one  independent 
variable,  and  the  values  of  r,  y,  &c.,  />,  q^  &c.,  -^  may  be  de- 
termined in  terms  of  t  and  of  their  values  x^,  ?/f  &c.,  jt?^, 
q    &LC.J  V    corresponding  to  the  value  t  of  t. 

Since  for  the  value  t  of  ^,  v^  becomes  a  given  function  of 
X,  y,  &c.,  it  is  evident  that  -ip,  must  be  the  same  function  of 
a;  ,  ?/  ,  &c. ;  and  also  that  p^,  q^,  &;c.  must  be  the  differen- 
tial coefficients  of  V^^  with  reference  to  x^ ,  y^,  (Sec. 

If  from  the  integrals  of  (39  d  and  40  d)  the  quantities 
Xr  ,  yr  ^-c.  are  all  eliminated,  and  the  value  of  i//  obtained,  this 
value  is  evidently  such  a  function  of  ^,  a;,  y,  &:,c.  that  if  ^,  x,  y, 
&c.  are  changed  to  ^,  '^^,  y^-,  ^-c.,  ^>  will  become  t/;  ;  but  by 
the  simple  change  of  ^  to  ^,  V^  must  become  the  same  function 
of  X-,  y,  &.C.  which  V  is  of  x  ,  y^ ,  &lc.  ;  that  is,  the  value  of 
ip  obtained  by  this  process  of  elimination  satisfies  the  problem. 

223.    Examples. 

1.  Integrate  the  linear  differential  equation  of  the 
first  order  J  involving  any  nnmher  of  independent  va- 
riables. 

Solution.     This  equation  may  be  written  in  the  form 

r  2>,  v^  +  XD,  V^  +  Fl>y  V  +  &c.  =  M,  (41  d) 

in  which  T,  JT,  Y,  &bc.,  M,  are  functions  of  t^  x,  y,  &/C.  and  V^. 


260  INTEGRAL    CALCULUS.  [b.  V.   CH.  XI. 

Equations  of  the  first  order. 

In  this  case  (36  d)  becomes 

R—Ts-\-Xp+Yg-\-&LC.  —  3I=0,  (42d) 

whence 

D,R=T,       DpR  —  X,  &c.  (43  d) 

s  D,R+p  D^R  +  &LC.=  T s+X p -\- &LC.  =  31,     (44  d) 

and  (39  d  and  40  d)  become 

1       D,x       Dty        „  D^-w 


The  fractions  in  the  first  line  of  (45 d)  do  not  involve  p,  q, 
&c.,  and  therefore  the  integrals  of  the  equations  in  this  line 
give  the  required  value  of  V^,  without  resorting  at  all  to  the 
second  line  of  (45  d). 

Corollary.  Whenever,  by  the  combination  of  the  equations 
in  the  first  line  of  (45  d),  a  number  of  equations  is  found  equal 
to  that  of  the  variables  v^,  x,  y,  &c.,  and  admitting  of  direct 
integration,  such  as 

D,  Z7=0,     A  F=r  0,  &,c.,         ^        (46  d) 

the  required  integrals  are 

C/z=  ?7^,     F=F^,&c.  (47  d) 

But  Z7  ,  F  ,  &/C.  are  functions  of  v^  ,  x^,  y^,  &lq,.,  from 
which,  by  the  elimination  of  a;  ,  y^ ,  &C.,  the  value  of  V^  may 
be  obtained  in  terms  of  C/. ,  F  ,  &c.  ;  and  the  required  yalue 
of  V'  is,  consequently,  the  same  function  of  U,  F,  &c. 

2.  Integrate  (41  d)  when  the  quantities  T,  X,  Y,  &c.  are 
functions  respectively  of  t,  x,  y,  &,c.,  each  function  involving 
but  one  variable. 


<§>  223.]    DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER.       261 
Equations  of  the  first  order. 

Solution.    In  this  case,  (45  d)  gives 

whence  the  values  of  x,  y,  &c.  are  determined  in  terms  of  t 
and  constants.  The  substitution  of  these  values  in  M  re- 
duces it  to  a  function  of  t^  which  may  be  denoted  by  iV, ,  and 
we  have  finally 


V 


H    N, 


-r-/:^.        («d) 


and   the  required   value   of  ^P  is  obtained    by  substituting  in 
(49  d)  the  values  of  x    ,  y    &-c.,  obtained  from  (48  d). 

The  values  of  a:  ,  y  ,  &c.  may  easily  be  derived  from  the 
values  of  x,  i/,  &i,c.  by  chauging  ^,  t,  x  ,  y  ,  &.C.  into  t,  t",  a:, 
y,  &c.  ;  for  the  values  of  x,  y,  &c.  belong  to  one  end  of  the 
interval  t — r,  in  the  same  way  in  which  z  ,  y  ,  &c.  belong  to 
the  other  end  of  the  same  interval,  so  that  t^  may  be  considered 
as  the  variable,  while  t  is  constant. 

3.  Integrate  the  equation 

t  D,rp  —  x  D,V^  =  0.  (50  d) 

Ans.     .p  =f.  V(.r^— <-+t2),     (51  d) 

in  which  y.  denotes  the  function  which  V^  is  of  x  when  t  be- 
comes T. 

4.  Integrate  the  equation 

X  D.V'  +  t  D,x  =  0. 

Ans.     V'=/(yj, 
wherey.  has  the  same  signification  as  in  (51  d). 


262  INTEGRAL    CALCULUS.  [b.  V.   CH.  XI. 

Equations  of  the  first  order. 

6.    Integrate  the  equation' 

.„.  ,=  (4)V.(v'> 

wherey.  has  the  same  signification  as  in  (51  d). 

6.  Integrate  tlie  equation 

atD,-^'-\-bxD:,^  —  n^.  (52  d) 

^7zs.^=  (^y-/[^(7)""].     (53d) 
wherey.  has  the  same  signification  as  in  (51  d). 

7.  Integrate  the  equation 

at""  D,y^  +  b  x''  DxV'  =  n  v>^  (54  d) 

r     1       ,    n{m-\)/    1  1   \-i-("«-i) 


where   y.  -  f  V  ± -^JtD  (  ^^ L_\"| 

wnere   v^^_/.  |^^,._^       ^  (A-1)  V''-^        r'^-'  )j 

andy.  has  the  same  signification  as  in  (51  d). 


(55  d) 

-(fc-i) 


8.  Integrate  the  equation 

t  DtV^  -}-y^  D^y^  -{-  X  z=z  0. 

where         x^  =:  x  cos.  log. yj  sin.  log.  —  , 

and  f.  has  the  same  signification  as  in  (51  d). 

9.  Integrate  the  equation 

t  (b+D,  '^)—x{a+Dt  V^)-\-rp(b  D^^p—a  Z?,  V^)=0.  (56  d) 


«§>  223.]    DIFFERENTIAL   EQ,UATIONS  OF  FIRST  ORDER.       263 


Equations  of  the  first  order. 


Solution.    In  this  case  (39(1  and  40  d)  give 

1        _    D,x     _      D,v^ 
b  V — X         t — a  ^  a  x — b  t 

Hence  •  i>,  i/'  +  6  Z),  z  -[-  a  =  0 

and  -qjD.^-^-xDiX-^-tzz^O, 

the  integrals  of  which  are 

^  —  ^^  +  b  {x—x^)  +  a  (t—r)  =  0 

V/2_^2  J^x^  —  X^-\-  t^  —  r^  —  0. 

Hence  _  ^2  _  ^2  _[_  ^,2  _  ^.2 

and  the  required   integral   is  obtained  by  the  elimination  of 
X    between  the  equations 

[2 V^+b{x-x^)+a{t-r)][h{x-Xr)+a{t-r)]z=x^-x2-\-t^-T^,  (57d) 

V^-/-  ^r+b  {x-x^)+a{t-r)=0,  (58  d) 

where/*,  has  the  same  signification  as  in  (51  d). 

Corollary.    The  integral  of  this  equation  is,  by  the  corollary 
to  the  first  example, 

^  +  bx-]-at  =v.(^'^  +  x^  +  t%  (59  d) 

in  which  9 .  is   an   arbitrary  function  to  be  determined  by  the 
condition  that 

f.x  +  hxJrar^^.lif.xf  +  x'^  +  ^l         (60  d) 
or  it  may  be  that   V  +  6x-|-«^    is  a  given    function   y    of 

V/2  _|_  x2  J^  t:2. 

10.    Integrate  the  equation 

at  Dt^  -{-b  xb^  -\-c  y  Dyyp+  &C.  =z  n  1/^.  (61  d) 


264  INTEGRAL    CALCULUS.  [b.   V.  CH.  XI. 


EqtiJitions  of  the  first  order. 


AtlS.      V  = 


in  which y.  is  the  function  which  V^  is  of  x^  y,  &c.  when  t  be- 
comes r. 


Corollary.     When 

a  =  6  =  c  =r  &c.  =  1,  (63  d) 

(62  d)  becomes 

so  that  -ip  is,  in  this  case,  a  homogeneous  function  of  the  nth 
degree,  of  ^,  x,  y,  &c. ,  and  (61  d)  becomes,  by  the  substitution 
of  (63  d),  a  proposition  applicable  to  such  functions. 

11.    Integrate  the  equation 

in  which  Z^  is  a  given  function  of  ^,  iT  a  given  function  of  t^, 
and  L  a  given  function  of  lt-\-h-{-^y-\-  ^-c. 

Solution.     Let         M=l,  +  lx  +  ly  +  &LC.,  (66  d) 

and  (39  d  and  40  d)  give,  in  this  case, 

L-^al,~     L-^al,  L+a  l^  ~       '       n    '     ^        ' 

Hence,  by  the  theory  of  proportions,  and  since 

D, M=D,  h+D,  h .  D,  x+Dy  ly .  B,  3/+&C.      (68  d) 

D,M    _B,.{l,-h)  _DM-ly)  B,^'      ^gj 

nL^aM  —  a{l,—h)    ^  a{lc-ly)  '        ^    '    ^         ^ 

where  n  is  the  number  of  the^  variables  t^  x,  y^  &c. 


<5>223.]      DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER.       265 

Equations  of  the  first  order. 


Whence,  since  L  is  a  function  of  iff, 

(70d) 

J  ]\lnL+am        ^  l—L  "^   I -I,  J  yj    n* 

where  ^^  =/.  (z^  ,  ?/^  ,  &c.),  (71  d) 

f  having  the  same  signification  as  in  (52  d) ;  and  the  required 
integral  is  obtained  by  the  elimination  q{  x^,  y  ,  &/C.  between 
the  equations  (70  d  and  71  d). 

12.    Integrate  the  equation  (65  d)  when 

L  =  m  M,  and  n  —  h  ^j.  (72  d) 

■ 

Ans.     The  equation  obtained  by  eliminating  Jf.  between 

6  a  a 

fM  \  m.  n-\-a      ^         ,M  \mn-\-a  (M 


■"  a  (73  d) 

V/--J,/-  =  ( J/ )         •  (74  d) 

wherey  is  the  function  which  ■^p  becomes  of/,,  Zy,  &c.,  when 
^  becomes  t. 

When  mn  -\-  a  =  0, 

the  integral  becomes 

(nl-M  Y-^r,   ,  I  M(l,-h\    ,     ,  3/(/,-/  ) 

13.    Integrate  the  equation  (65  d),   when  L  is  any  given 
function  of  the  variables  ^,  z,  y,  &,c.,  and  V^. 

Solution.    In  this  case,  the  first  member  of  (69  d)  must  be 
omitted,  but  the  other  member€  give  the  values  of  all  the  va- 
23 


266  INTEGRAL    CALCULUS.  [b.  V.   CH.  XI, 

Equations  of  the  first  order. 

riables  expressed  in  terms  of  any  two  of  them,  and,  therefore, 
the  value  of  L  expressed  in  terms  of  these  two,  which  two 
may,  for  instance,  be  '^  and  t,  and  the  equation 

will  be  a  differential  equation  of  the  first  order  between  two 
variables,  and  may  admit  of  easy  integration. 

Corollary.  This  method  may  be  applied  in  any  case, 
in  which  all  the  integrals  hut  one  of  a  system  of  diffe- 
rential equations  of  the  first  order  have  been  obtained^ 
and  the  final  integral  will, depend  only  upon  the  integra- 
tion of  a  differential  equation  of  the  first  order  between 
two  variables. 

14.  Integrate  the  equation  (65  d),  when  Z*  is  a  given  func- 
tion oi  l^+lx  +  ly  +  &c.  +  l^  ,  and 

L-\-al 


u— 


^^K 


^  .  (77  d) 


Ans.    The  equation  obtained   by  eliminating  a:^,  y^,  6lc. 
between  the  equations 

aJV         7_7  7_7  7 7 

T     .r  T     V  I-        xp 

T  -^T  T 

where  V^    has  the  same  signification  as  in  (71  d),  and 

iVrr=  r^^  ^- ^,  (79  d) 

J  31    nL+a31  ^        ^ 

J[f=?,  +  Z^  +  /,  +  &c.  +  7^,  (80d) 

and  n  is  the  number  of  the  viriables  t,  z,  y,  &c.,  and  V. 


<§>  223.]    DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER.       267 


Equations  of  the  first  order. 


15.  Integrate  the  preceding  example  when 

l  =  mt,     M—L.  (81  d) 

Ans.    The  equation,  obtained   by  eliminating  L^  between 

(  —     )nm^a  =  ^  =  —  ,  (82  d 

where  i//    is  the  same  as  in  (71  d). 

When 
the  integral  is 


m  »  -)-  ^  =  ^> 

(83  d) 

t  —  ^         nt  —  L 

T  V^      ~WT L 

X 

(84  d) 

16.  Integrate  the  equation 

TJ^X-\-  Y+&LC.  =  0,  (85d) 

in  which  T  is  a  function  of  t  and  Z)^  V,  JIT  a  function  of  x 
and  D^  v^ ,  Y  a  function  of  y  and  Dy  ip ,  &/C. 

Solution.    In  this  case  (39  d  and  40  d)  become 
1  D,x  dt  y 


=z  &C. 


D,T        D^X        D^Y  (86  d) 

_    -P^p  ^      Ay  _o A  ^ 

~    AX        A  ^      ""    5A2^+i^AJR-?A^-+-&,c. 

which  give  the  equations 

A^  A^  + A^  Ai'^o 

A  i'.Ay  +  A  ^-A^^O,  d6C.,    (87  d) 

the  integrals  of  which  are,  by  denoting  the  values  which  X^ 
Y,  &LC.  have  when  x,  y,  6lc.^p,  q,  &:,c.  become  a;  ,  y^ ,  &:-c. 
by  X^,  Y^,  &LC., 

X=:X.   Y=  F,&c.  (88d) 


268  INTEGRAL    CALCULUS.  [b.  V.  CH.  XI. 

Equations  of  the  first  order. 

Hence,  by  (85  d), 

rz=  J\.  (89  d) 

These  equations  give  s,  p,  q,  &c.   in   terms   respectively  of 
t,  X,  y,  &/C.,  which  substiluled  in  (66  d)  give,  by  integration, 

These  equations  give  r,  ?/,  &c.  in  terms  of  t,  which,  substi- 
tuted in  (SG  d)  give 

where  V  is  the  same  as  in  (71  d).  The  integral  of  the  given 
equation  is,  finally,  the  equation  obtained  by  eliminating  x^, 
y  ,  &c.  between  (9J  d  and  91  d).  In  making  this  elimina- 
tion, it  is  to  be  observed  that 

Pt  =  -0.    "^r-    <l^  =  ^y  •V'^.&c.  (92d) 


17.  Integrate  (85 d),  when  T  is   a  function  of  D^  ^,  Xof 

Solution.    In  this  case  (86  d  )  gives 

P=Pr'     g'  =  g^,&c.  (93  d) 

y-y,==  J^;J(<-^)>&'C.,  (94  d) 


V; 


_,^^  i^.i±PA^^^^±^  (,_.).      (95  d) 


«§>  223.]    DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER.       > 


269 


Equations  of  the  first  order. 


Hence,  if  V'^  is  used  as  in  (71  d),  and  if 

T 
T 

the   required  integral   is  obtained  by  eliminating  s,  p,  q,  &;c. 
between  the  given  equation  and  the  equations 

^sV.T  +  ,I>,^X^j^Y^^_^^^  (97  d) 


18.  Integrate  the  equation 

Ans.    The  equation  obtained  by  eliminating  p  between  the 
equations 

V'   =/.    [X   +   2   «  p    (^  —  T)]    _   «  p5    (^   _   T) 

and  P^f-'Vx  +  'Hap  [t  —  r)l 

where  /  and/'  are  used  as  in  (71  d  and  95  d). 

19.  Integrate  the  equation 

{DrPf  =  b{Dj:^y~.  (99  d) 

Ans.  ^=f.[x  +  {t-^)^b],     (le) 
where  /is  used  as  in  (51  d). 
23* 


270  INTEGRAL    CALCULUS.  [b.   V.  CH.   XL 

Equations  of  the  first  order. 

20.    Integrate  the  equation 

/e  =  0,  (2e) 

where  7?  is  a  function  of  T,  X,  F,  &c.,  which  have  the  same 
signification  as  in  (85 d). 

Solution.  It  may  easily  be  shown  that  the  equations  (87  d, 
88  d  and  89  d)  are  applicable  to  this  case.  Hence  the  values 
of  s,  p,  q,  &c.  may  be  found  in  terms  respectively  of  t,  x,  y, 
&c.,  and  these  values,  substituted  in  (32 d),  reduce  the  suc- 
cessive terms  to  functions  respectively  of  f,  x,  y,  &c.  ■  The 
integral  of  (32  d)  gives,  therefore, 

^.,^=j\s+f^y+f^^^,  +  6.o..  (3e) 

and  the  value  of  V',  obtained  by  eliminating^  ,  q  ,  &lc.  be- 
tween (2  e,  3  e,  and  92  d),  satisfies  (2e).  The  values  of  x  , 
7/  ,  &c.  are  finally  eliminated  by  means  of  the  integrals  of  the 
upper  line  of  (39  d),  which  is  freed  from  s,p,  q,  &c.  by  means 
of  (88  d  and  89 d).* 

The  functions  D^R,  Dx  -K,  ^c,,  are  functions  of  T,  X^ 
&c.,  and  therefore  by  (88  d  and  89  d)  they  are  constant,  so 
that  the  integrals  of  the  upper  line  of  (39  d)  become 

(4e) 
and  the  required  integral  is  therefore  the  result  of  the  elimina- 
tion of  a;   ,  y   ,  &/C.  between  the  equations  obtained  from  (3  e 

*  Note.  This  last  process,  which  is  necessary  in  order  that  \fj  may 
become  a  given  function  of  x,  y,  &c.  when  t  becomes  t,  is  neglected 
in  the  ordinary  solution  of  this  question  given  in  (Lacroix,  Calc.  DifF. 
et  Int.,  2d  ed.,  Vol.  I,  p.  572). 


<5>  223.]    DIFFERENTIAL  EQUATIONS  OF  FIRST  ORDER.       271 
Equations  of  the  first  order. 

and  4  e)    by  the  substitution  of  (92  d)  and  of  the  value  of  s 
obtained  from  (2  e)  by  changing  T,  A",  Y,  &c.  to   T  ^  X  ^ 
F  ,  &c. 

21.  Integrate  the  preceding  example  when  T*,  X,  Y,  &;c. 
are  the  same  as  in  example  17. 

Ans.  The  integral  in  the  equation  obtained  by  the  elimina- 
tion of  X  ,  1/  ,  ^c.  and  5^ ,  between  the  equations  obtained 
from 

V/  =  V^^  +  ^  (^— ^)  +  P.  (•^— ^r)  +  Q't  ilZ—^r)  +^^'      (5  e) 


T  T  T 


and  ^r  —  ^>  (^®) 

by  the  substitution  of  (92  d). 

22.    Integrate  example  20,  when 

T=.T'D,^,     X=X'  D,V^,&LC.  (8e) 

where  T',  X\  Y',  &c.  are  functions,  respectively,  of  t,  x,  y, 
&c. 

Ans.     The  integral  is  the  equation  obtained  by  the  elimina- 
tion of  5    i  '^r  i  y^i  &c.  between  the  equations  obtained  from 

r       't 

(9e) 

and  -R^  ==  0,  (lie) 

by  the  substitution  of  (92  d). 


272  INTEGRAL    CALCULUS.  [b.   V.  CH.  XI. 


Equations  of  the  first  order. 


23.    Integrate  example  20,  when 

T=  T'  {D,rp)\     Xz=  X'(l>,t/.)'"&c.,      (12  e) 
where  T',  JC',  &c.  are  the  same  as  in  the  preceding  example. 

Ans.    The  integral  is  the  equation  obtained  by  the  elimina- 
tion of  5   ,  X   ,  v/   ,  &c.  between   the  equations  obtained  from 


us-wt;^-'^         v^' 


71—1  _  I 

J     ^ 


= \ r^    — ^- —  —  &LZ.      (14  e) 

«/  X 


m 


p;-^  X;  -^  D.  R^  ^  ^    s/X' 


and  (He)  by  the  substitution  of  (92  d). 

24.    To  integrate  the  equation 

jR  =  0,  (15e) 

when  i?  is  a  function  of  t,  x,  y,  &-c.,  X?, ,  V^ ,  Z>^  V ,  &c.  and  (p 

where 

g)=z^I>,t/^  +  a?X>,  V^  +  «Sz>c.  —  V^.  (16 e) 

Solution.     If  s,  ^,  q,  &c.  have  the  same  signification  as  in 
§222,  (16  e)  gives 

g)  z=  ^  5  +  ojp -}-  &c.  —  V',  (I'^e) 

and  if  the  differentials  are  taken,  as  if  5,^,  q,  ^-c.  were  the 
independent  variables  of  which  t,  x,  &c.  are  functions,  we 
have 

D,cp  =  t-^s  D,  t-\-p  A  X+&C.— D,  V^  A  t—D,^  J9;a;— &c. 
==<,  (18  e) 


§  224.]     DIFFERENTIAL  EQUATIONS   OF   F I  LIST  ORDER.       273 
Simultaneous  equations. 


and,  in  the  same  way, 

Dp(p  =  x,     X>,  (jp  =  y,  &.C.  ;  (19  e) 

that  is,  t,  r,  ?/,  6lc.  are  the  difFerential  coefficients  of  q)  rela- 
tively to  5,  }),  q,&LQ>.,  and,  therefore,  the  integral  of  {\o  e)  may 
he  obtained  as  if  cp  were  the  unknown  function,  s,  p,  q,  Sfc. 
the  independent  variables,  and  t,  x,  y,  <^'c.  the  respective  diffe- 
rential coefficients. 

224.  When  the  required  function  -^  is  dependent 
upon  several  variables,  there  may  be  several  given  equa- 
tions between  its  differential  coefficients,  and  the  solu- 
tion is  possible,  provided  the  number  of  equations  does 
not  exceed  the  number  of  variables.  In  this  case  of 
several  simultaneous  equations^  as  many  differential  co- 
efficients maybe  eliminated  as  the  number  of  equations 
exceed  unity  ;  and  the  resulting  equation  may  be  inte- 
grated by  the  preceding  methods.  It  is  to  be  observed 
that,  in  the  integration  of  this  equation,  those  variables 
ma}''  be  regarded  as  constant,  of  which  the  correspond- 
ing differential  coefficients  have  been  eliminated.  The 
relation  of  the  required  function  to  the  variables,  which 
have  been  thus  regarded  as  constant,  is  determined  by 
substitution  in  the  given  equations  of  the  result  of  the 
integration.  The  limits  of  this  volume  do  not,  how- 
ever, permit  any  examples  of  this  process. 


274  INTEGItAL    CALCULUS.  [b.    V.    CH.   XII. 

E(]uatiuns  of  ll)e  second  order. 


CHAPTER    XII. 

INTEGRATION     OF     DIFFERENTIAL    EQ,UAT10NS     OF     THE 
SECOND    ORDER. 

225.  Problem.  To  find  a  function  ^  of  two  varia- 
bles, X  arid  t,  which  satisfies  a  given  differential  equation 
of  the  second  order ^  and  which  becomes  a  given  function 
of  X  for  a  given  function  r  of  t,  and  its  first  differential 
coefficient  D^  V  ,  taken  relatively  to  t,  becomes  another 
give?i  function  of  x  for  the  same  value  of  t. 

Solution.     Let 

and  let  the  given  equation,  by  the  substitution  of  these  values, 
become 

R  —  0.  (21  e) 

If  D'  denotes  the  differential  coefficient  of  a  function  taken 
relatively  to  either  of  the  given  functions  of  x,  we  have 

D'-^^pD'x,     Dp  =  iDx,     D's  =  Q.D'x.     (22e) 

Although  X  is  independent  of  t,  it  may  be  assumed  to  be  an 
arbitrary  function  of  t,  and  in  this  hypothesis  V^,  p,  5,  &c.  will 
become  functions  of  t,  and  will  give 

DtV'  =  s  -{-p  D,x,  (23  e) 

D,y^  =  Q  +  ^^D,x 


(24  e) 


^  225.]    DIFFERENTIAL  EQUATIONS  OF  SECOND  ORDER.    275 

Equations  of  the  second  order. 

The  differential  coefficients  of  (24  e)  are 

D'D.s   =z  D'o-^  D'Q.D,x  +  o  D' D,x,       ^ 


and  those  of  (22  e)  give 

S    (26  e) 


D'  DtP  =  D^  I  D'  x-\.  i  D'  D,x 


D'  D,s  =  D,Q  D'  X  -\-Q  D'  D,x; 

which,  substituted  in  (25  e),  give 

D'q  =  D.c.D'  X  —  Df^  X,  D',-  ,  (27e) 

D^  =  D,Q  .  D,  X  —  D,  X  .  D'  Q 

=  {D^Q  —  D/i.D.x)  D'x-\-{D,xfD'  ^.    (28e) 

If  the  values  of  D' M>,  D' p,  D'  s,  D'q  and  D' a  (22  e,  27  e, 
28  e)  are  substituted  in 

D'  R=0,  (29  e) 

the  resulting  equation  contains  the  two  arbitrary  and  indepen- 
dent elements  D'  x  and  D'  '^ ,  the  coefficients  of  which,  being 
put  equal  to  zero,  give  the  two  equations 

D^R{D,x)^  —  DR.D,x-]-D,R  =  0,         (30  e) 

D^R{D,Q—Dic.D,x)  -{-D^RD,c  +  QD,R  +  iD^R 
+  pD^R  +  D^R  =  0.  (31  e) 

Whenever,  from  a  judicious  combination  of  the  equations 
(21  e,  23  e,  24  e,  30  e  and  31  e),  three  equations  can  be  found 
capable  of  integration,  the  elimination  of  ? ,  o^  and  ;  between 
the  three  integrals  of  these  equations  and  the  equation  (21  e) 
will  give  two  equations  between  p,  s,  V,  x,  \ »  ^^ ,  5^ ,  p^ ,  o^ 
and  z  .  In  the  two  equations  thus  obtained,  D,  V  and  Dx  i// 
may  be  substituted  for  s  and  p,  Dz^  xp^  for  p^ ,  Dl   '^P^  for  t^ , 


276  INTEGRAL    CALCULUS.  [b.  V.  CH.  XII. 


Equations  of  the  second  order. 


Di-  s  for  e  ,  and  two  functions  of  x  for  -^p  and  >•  corres- 
ponding  to  the  given  functions  of  a-,  which  ■^P  and  .s  become 
when  t  becomes  r.  Between  tlie  two  equations  thus  obtained, 
X  may  be  eliminated,  and  tlie  resulting  equation  is  a  differen- 
tial equation  of  tlie  first  order,  and  its  integral,  obtained  by 
the  methods  of  the  preceding  chapter,  is  the  required  inte- 
gral. ^ 

This  process  is  precisely  similar  to  that  of  §  222,  and  is 
derived  from  the  same  principles. 

226.  Corollary.  The  two  given  functions  of  x  are  wholly 
arbitrary,  and  may  be  altogether  independent  of  each  other. 
They  involve,  therefore,  in  the  general  value  of  V^,  two  inde- 
pendent and  arbitrary  functions  of  V,  x  and  t,  and  which  may 
be  independent,  not  merely  in  reference  to  the  nature  of  the 
functional  operations  themselves,  but  in  regard  to  the  variables, 
that  is,  to  the  cotnbinations  of  V',  x  and  t,  upon  which  they 
depend.  This  variable  or  combination  is  represented  by  x^  in 
the  preceding  section.  There  must,  therefore,  in  general,  be 
two  different  values  of  x  ,  each  of  which  will  give  a  different 
equation  of  the  first  order,  the  integral  of  either  of  which  leads 
to  the  required  value  of  ¥'.  But  instead  of  integrating  the  two 
equations  of  this  first  order  independently  of  each  other,  it  will 
be  found  much  easier  to  integrate  either  of  the  equations  ob- 
tained from  them  by  the  elimination  of  D^V^  or  DtV^. 

227.  Scholium.  There  are  many  cases  in  which  it  is  ex- 
pedient to  transform  the  given  equation,  before  applying  this 
process  of  integration  ;  and  some  of  them  will  be  considered 
among  the  examples. 

Whenever  the  three  required  integrals  cannot  be  obtained, 
the  preceding  process  is  inapplicable,  although  the  given  equa- 
tion may  sometimes  admit  of  integration  in  these  cases,  by 
means  of  analytical  artifices. 


'5>228.]    DIFFERENTIAL  E(^UATIONS  OF  SECOND  ORDER.    277 
Equations  of  the  second  order. 

228.     Examples. 

1.    Integrate  the  equation 

2>;  V^-|_  (a+6)  Dl,rp  +  abDl^p  =  P,         (32  e) 
in  which  P  is  a  given  function  of  x  and  t. 

Solution.     In  this  case  the  equation  (21  e)  is 

a  _(-  (a  _j-  6)  ?  +  a  6  a  =  P,  (33e) 

and  (29  e)  is 

D'o^  (a  +  b)  D'  Q  +  ab  D'^^  =  D^  P.  D' x.     (34  e) 

Hence,  by  the  substitution  of  (27  e  and  28  e),  the  coefficient 
of  D'  i  placed  equal  to  zero,  is 

{D,x)^-'(a  +  b)  D,x-{-abz=0;  (35  e) 

whence 

DtX  =z  a  or  =:b,  (36  e) 

x  —  x^=z  a  {t—r)  0Y  =  b  {t  —  r).         (37 e) 
The  first  of  these  two  values  reduces  (23  e  and  24  e)  to 


>      (38  e) 

D,s  +  b  D,p  =  P,  (39  e) 

in  which  P  may  by  (37  e)  be  reduced  to  a  function  of  t^  and 
denoted  by  P^•,  hence 

5  -  ^r  +  ^  {P-Vr)  =fi  ^r.  (40  e) 

In  the  same  way,  if  s'^ ,  p^  and  Pi  denote  the  corresponding 
values  when  the  second  equations  in  (36  e  and  37  e)  are  em- 
ployed, we  find 

5-<  +  a(p-p;)=/;P;.  (41  e) 

24 


278  INTEGRAL  CALCULUS.      [b.  V.  CH.  XIL 

Equations  of  the  second  order. 

These  two  equations  become,  by  the  substitution  o(  f.x  for  the 
values  of  v^  and  D^  V^  when  t  is  r, 

D:rp+b  D,rp-/o(x-at+aT)-bD^f{x-at+ar)=Q,  (42e) 
Dt  tp+a  Z>, V-/o  (x-b  t+b  r)-a  D,f{x-b  t+h  '^)=Q',    (43  e) 

in  which  Q  and  Q'  are  the  values  which  the  second  members 
of  (40  e  and  41  e)  acquire  by  the  substitution  for  x  of  its 
value  given  by  (37  e).  The  combination  of  these  two  equa- 
tions gives 

{a — b)D,  ^ — afo  (x — at-\-a  '^)-\-bfo  {x—b  t-\-b  r) 
-abD^f.(x-at+ar)+abD,f{x-bt+br)=aQ-b  Q',  (44  e) 

(a—b)  D^  V'-L/o  (x— «  t+a  r)—f,  (x—b  t+b  r) 

+6  D,f  {x—a  t+ar)—af(x—b  t+b  r)=Q'—Q^    (45  e) 

the  integral  of  which  is 

(a-b)  y^+f.fo  {x-a  t+a  r)  -f^f,  (x-b  t+b  r) 

+bf{x-at+ar)-af.(x-at+a'^)=f^(a  Q-b  Q').    (46e) 

2.  Integrate  the  preceding  example  when 

P  =  tx. 

Ans.     The  equation  (46  e)  when  its  second  member  be- 
comes 

i  (a-6)  {t-rf  [i  X  (<+2  T)  -T-V  (a+h)  {t+3  t)  (<-t)]. 

3.  Integrate  the  equation 

t^-  DlV^  +  2tx  Dl,-^  +  x^  Dlrp  =  P, 
in  which  P  is  a  given  function  of  x  and  t. 


<5>  228.]    DIFFERENTIAL  EQUATIONS  OF  SECOND  ORDER.    279 


Equations  of  the  second  order. 


Am.  v=/.  — +  (/«•  y-  +  7  D.f.  —  )  (<-,)+  Q, 

t  X 

in  which  P^  is  the  value  of  P  when  x  is  changed  to  — -,  Q  is 
the  value  of 

J  X    J  r       t^    ' 

X   T^      . 

when is  substituted  for  x   . 

t  ^ 

4.    Integrate  the  equation 

n  Mj  (^^^) 

in  which  P  is  a  function  of  ^±^ 

Solution.     In  this  case  (30  e)  becomes  by  (20  e), 
p^  {D,  2;)2  +  2  p  5  i>,  a:  +  s2  =  0, 

5 

whence  DtX^z ,  (48  e) 

n  ' 

and  by  (23  e,  24  e  and  27  e), 

A  V^  =  0,  (49  e) 

Dtp  =  Q -,  (50  e) 

p 

D,s=o--  =  --'^+Pp=-D,p  +  Pp.    (51  e) 

P  p  p-        *  P 

Hence 

D,.~~P  and       ll>,.-=l;         (52e) 

p  F         p  ^        ' 


280  INTEGRAL    CALCULUS.  [b.   V.   CH.  XII. 


Equations  of  the  second  order. 


S 
and  since  P  is  a  function  of—,  the  integral  of  (52  e)  may  be 

directly  obtained,   and   gives    —  in  terms  of  t,  and  this  value 

substituted  in  (48  e)  gives 

''  —  \  =  -fr     J*  (^^^) 

whence,  by  (49  e), 

^  =  ^r=/-^5  (54  e) 

and  the  required  integral  is  obtained  by  the  elimination  of  x 
between  (53  e  and  54  e). 

5.  Integrate  (47  e)  when  the  second  member  is  zero. 

Ans.     x  =  F.rp—\,     ^       (t  —  r),  (55  e) 

in  which  JFis  the  inverse  function  of  f,  and  f  is  the  differen- 
tial coefficient  of  P  taken  with  respect  to  its  variable. 

6.  Integrate  the  equation 

D^y^.Dlv^  —  (Dl^  ^f  =  0.  (56  e) 

Solution.     In  this  case  (21  e)  is 

.     a?_^2_o,  ,  (37  e) 

and  (30  e)  is 

s^  {D,xf  +  2QD,x  +  a  =  0, 

whence  q'^  {D,  xf  +  2  Q  o  D^  x  +  o^  =  0 

o 

and  (23  e  and  24  e)  give 

Z)j  5  =  0,       S  =z  Sr 

Ap=0,    j)  =  p^ 
Dt^P  =  s^+p^D^x 


«§)  228.]    DIFFERENTIAL  EQUATIONS  OF  SECOND  ORDER.    281 


Equations  of  the  second  order. 


and  (31  e)  gives 


whence 

T  T 

and  the  required  integral  is  the  result  of  the  elimination  of 
x^  between  the  two  equations 

^-^^=f^{t-r),.  (58e) 

^  -/.  x^  =fo  .x^{t-  r)  +/:  x^  {X  -  x^) ,       (59  e) 

in  which  the  accents  denote  the  successive  differential  coef- 
ficients of  the  functions. 

7.  Integrate  the  equation 

when  the  value  of  V^  becomes  f  x .  f^  x  when  t  becomes  r,  and 
the  value  of  Dt  H'  becomes 

f  x.f^x—fx.flx, 

in  which  f  andy^  are  the  differential  coefficients  of/*  andyo. 

Ans,     ^  =  /  (z  +  ^  —  t)  .  /j,  (a;  —.  ^  +  T) . 

8.  Integrate  the  equation 

D]^-'DI^  —  -D:V^  =  0.  (60e) 

Solution.  In  this  case,  the  general  form  of  solution  is  in- 
applicable without  previous  transformation.  For  this  purpose, 
by  putting 

V^'  =  D^  V,  (61  e) 

24* 


282  INTEGRAL    CALCULUS.  [b.   V.  CH.  XII. 


Equations  of  the  second  order. 


the  differential  coefficient  of  (60  e)  relatively  to  t  is 

D'ln^'  —  DlH^'—'^^D.^'  +  '^^^rp'^O.  (62e) 

The  integral  of  (62  e)  may  be  found  by  the  general  process, 
which  gives 

Axzrril,      2;  — x^=±(<  — t),  (63  e) 

■^  2  5         2  V 

As  =  <^±e  =  rh  Ap  +  -f-— ^,  (64  e) 

D.pzzzQ:^^,  (65  e) 

Z>,  V^'  =  5  rh  i?,  (66  e) 
and  the  remainder,  after  subtracting  (66  e),  divided  by  t'^  from 


(64  e)  divided  by  ty  is 


—  it 


the  integral  of  which  is 

The  sum  of  the  equations  involved  in  (68  e)  is 

2D^       f,(x-t+r)-^f,{x-\-t-r)  ^  f;{x+t-r)-f;{x-t+r) 

^  T  T 

_^y_/,  (.-,+.)+/.(.+,_.) ^    (69e) 

in  which /j  x,  and  f^  x  denote  the  values  of  V^'  and  i^^V^' when 
t  is  T,  and  /q  x  is  the  differential  coefficient  of  y,  2  relatively 

^0  X. 


<§>  228.]    DIFFERENTIAL  EQUATIONS  OF  SECOND  ORDER.    283 


Equations  of  the  second  order. 


The  integral  of  (69  e)  is 
2  TV'' 


-FA^  +  t-r)-F,i.-t  +  .)^  (69  e') 

in  which  F^x  and  F^x  are  the   integrals  o{  f^x  and  y^  x 
relatively  to  x.     But  it  follows  from  (60  e,  61  e  and  63  e)  that 

fo^=fi^  (70  e) 

f,x=f:x  +  jf,x,  (71c) 

whence  (69  e')  becomes 

2  T  2>,  v^ 

+  /o(^  +  <— )+/o(^-«  +  -) 

and  V'  is  obtained  from  the  integral  of  (72  e).  ^ 

9.    Integrate  the  equation 

PDf  ^-\-S  Dl^^  -f  rDjV'rzrO,         (73e) 
in  which  P,  S,  and  T  are  functions  of  D^  r// ,  and  D   yj. 

Solution.     Let 

<P  z=:  s  t  -\-p  X  —  ^,  (74 e) 

in  which  s  and  p  have  the  same  signification  as  in  (20  e). 

And  if  the  differentials  are  taken  as  if  s  andp  are  the  inde- 
pendent variables,  of  which  t  and  x  are  functions,  we  have,  as 
in  (18  e  and  19  e), 

D,(p=t,     Dpcp=zx.  (75  e) 


284  INTEGRAL    CALCULUS.  [b.  V.  CH.  XII. 


Integration  of  equation  of  surface  of  minimum  extent. 


The  differentials  of  (75  e),  taken  as  if  t  and  x  are  the  inde- 
pendent variables,  while  the  first  members  are  expressed  in 
terras  of  p  and  s,  are 

{i=zD,D,cp=D].,cpD,s  +  DlcfD,p  )    ^      ^^ 

and  (76  e)  and  (77  e)  give 

Dl^V^  =  D,p=  D^s=z^MDl,(p  \  (78  e) 

in  which 

^={Dl,<pf-D]<p.Dl<p,  (79  e) 

and  the  substitution  of  (78  e)  in  (74  e)  gives 

PDl(p—SDl,cp+TD',cp  =  0,  •  (80e) 
which  may  be  integrated  as  if  p  and  s  were  the  independent 
variables. 

10.  Integrate  the  equation  of  the  surface  of  mini- 
mum extent. 

Solution.  By  changing,  in  (959),  x,  i/  and  z  to  t,  x  and  V, 
to  correspond  to  the  notation  of  this  section,  that  equation  be- 
comes 

(l+(^.V^)')i^?V>-2A^.-D.V'D^,V+(l+(AV)2)D^Vr=0. 

(81  e) 
By  the  substitution  of  the  preceding  example  this  equation  be- 
comes 
(l+p2)  Dlcp  +  2  p  s  Dl.  (p  +  (I  +  s^)  D',  9=0,  (82  e) 


4>  228.]    DIFFERENTIAL  EQUATIONS  OF  SECOND  ORDER.    285 
Integration  of  equation  of  surface  of  minimum  extent. 

which  cannot  be  integrated   by  the  direct  application  of  the 
general  process.     If^  however,  we  put 

cp'  =  D,cp,  (83  e) 

the  differential  coefficient  of  (82 e),  relatively  to  s,  is 

(1  +p^)  Dl  (p'  +  2  ps  Dl^ip'  +  (1  +  s2)  Dl  <p' 

+  2pD^,cp'-{-2s  D,cp'  z=zO.  (84e) 

The  general  process  applied  to  this  case  gives,  by  putting 

o  ■=Dlcp',  Q  =  Dl,cp',       i  =  Dlco',    ] 

(1  +s2)  {D,pf  —  2p  s  D,p  +  (1  +i>^)  -  0,    (86  e) 

the  integral  of  which,  found  by  the  process  of  Ex.   27  of 
§220,  is 

.-S;--^C--(f^')'].'-> 

in  which  5^  and  p    should  be  accented  when  the  lower  sign  is 
used. 

Instead  of  proceeding  with  the  direct  process,  we  may  put 
,n=P^^^,     n=P^,  (88e) 

5 5^  S  S^ 

T  T 

and  (87  e)  gives 

^  =  ws  +  V  (— 1— w^)  =  7is— V(— 1— «^),  (89 e) 
(l+s2)m2— 2psm  +  (l+p2)  =  0  | 

(l+s2)n2  — 2psw+(l+/?2)=0,  )    (^^®) 

Dr,p  =zm  D^s,     D^p  =  n  D^s,  (92e) 


286  INTEGRAL    CALCULUS.  [b.  V.    CH.  Xll. 

Integration  of  equation  of  surface  of  minimum  extent. 

D,,  cp'  =  {m  Dp  cp'  +  D,  <p')  D„  s,  (93  e) 

ni.n^p'  =[mn  Dlcp'  +  (m  +  n)  Dl.cp'  +  Dlcp']  D^^s.D^s 

+  Dpcp'  D^s  +  (nD,  <p'  +  D,  cp')  Z>L.  s 
=  —  [(I  +p^)Dlcp'  -\.2ps  Dip  <?>'  +  (1  +  s2)  Dl  cp'] 

l±f (2pDpCp'-{-2s  D,cp'){l  +5^) 

4  V  (-l-w^)  V  (-1-71^)        4  V  (—  1  —  wi"2}  V  (  —  1  —  ^2) 
=  0.  (94  e) 

Hence  we  find  by  integration, 

cp'  =z  F.m  —  F.m  -^  -\-f.  n,  (95  e) 

in  which  f.  n  is  the  function  which  <p'  becomes  when  m  be- 
comes m^,  and  Dj^f-'ni  is  the  function  which  D^  cp'  becomes 
for  all  values  of  n.  The  equations  (75  e  and  83  e)  give  the 
values  of  yj ,  t  and  x  in  terms  of  7?i  and  n.  But  it  may  be  ob- 
served that  t  is  the  same  as  cp'. 


^  230.]  PARTICULAR    SOLUTIONS.  287 

Particular  solutions. 


CHAPTER   XIII. 

PARTICULAR  SOLUTIONS  OF  DIFFERENTIAL  EQ,UATIONS. 

229.  In  addition  to  the  integral  of  a  differential 
equation,  there  are  particular  solutions^  which  are  not 
included  in  the  general  forms  of  the  integral. 

230.  Problem.  To  find  a  particular  solution  of  a 
differential  equation  of  the  first  order  between  two  va- 
riables. 

Solution.     Let  x  and  t  be  the  variables,  and  let 

R  =  0  (96  e) 

be  the  equation,  and  let 

X=0  (97  e) 

be  the  required  solution  which  is  supposed  not  to  be  included 
in  the  general  form  of  the  integral  represented  by 

V=0.  (98  e) 

If,  however,  (97  e)  were  a  case  of  (98  e)  corresponding  to 
the  value  x '  of  the  arbitrary  constant  x  involved  in  (98  e), 
and  if  we  put 

h  =  x^  —  x^,  (99e) 

the  difference  between  the  values  of  x  derived  from  (97 e  and 
98  e)  must  vanish  with  h,  so  that  where  h  is  an  infinitesimal, 
if  x'  denotes  the  value  of  x  derived  from  (98  e),  and  x  its  value 
from  (99  e),  we  may  put 

x' —  a;  =  JT' A",  (If) 


288  INTEGRAL    CALCULUS.  [b.  V.  CH.  XIII. 

Particular  solutions. 

in  which  X'  is  a  function  of  x  and  t  different  from  zero.  If 
^  is  the  value  of  Dt  x  given  by  (97  e),  and  p'  its  value  given 
by  (98  e),  we  shall  also  h^  \e,  when  x'  —  x  is  an  infinitesimal, 

j9'— ;;  =  P'(r' _  2:)"'=^ P';r ""/*'"",  (2  f) 

in  which  P'  is  a  function  of  x  and  t  different  from  zero. 

But  the  differential  of  (If)  gives 

P'-P  =  n,X',h\  (3f) 

Whence  we  must  have,  if  (97  e)  is  a  case  of  (98  e), 

P  X'  "^  h'^ "  z=  i>,  X.  h\  (4  f ) 

But  if  ?7i  is  less  than  unity,  A"*"  will  be  infinitely  greater  than 
A",  and  the  equation  (4  f )  becomes 

P'  X"^=-  0,  (5f) 

which  is  impossible,  so  that  in  this  case  (4  f )  cannot  be  satis- 
fied, and  (97  e)  is  not  a  case  of  (98  e),  and  is  consequently  a 
particular  solution. 

If  »»  had  been  unity,  (4  f )  would  have  been  reduced  to 

P'  X'  =  D,  X',  (6  f ) 

which  is  easily  satisfied. 

If  m  were  greater  than  unity,  (4  f )  becomes 

D,  X'  =  0,     X'  =  constant,  (7  f ) 

so  that  a  particular  solution  is  only  indicated  by  the  condition 
that  m  is  less  than  unity. 

The  differential  of  (2  f )  gives 

X>,,  p'  =.mP'  {x'  —  xY-\  (8  f ) 

which,  when  x'  differs  infinitely  little  from  x  and  m  is  less  than 

unity,  gives 

D.p^-'^i;  (9f) 

that  is,  DxP  is  a  fraction  whose  denominator  is  zero. 


<5.  232.]  PARTICULAR  SOLUTIONS.  289 


Particular  solutions. 


The  diflferentialion  of  (96  e)  relatively  to  x  gives,  by  substi- 
tuting J)  for  D^  X, 

D^  R  .  D^p  +  i>,  72  =:  0,  (10  f ) 

''■^  =  -d;r-  (i»o 

Whence  by  (9  f ), 

i>^R=0,  (12  f) 

provided  the  numerator  cannot  become  infinity,  which  will 
be  the  case  when  (96  e)  is  free  from  radicals  and  fractions. 
This  equation  (12  f)  corresponds  to  the  particular  solution, 
and  leads  to  the  particular  solution  by  the  elimination  of  p  be- 
tween it  and  the  given  equation  (96  e). 

231.    Corollary.  A  similar  method  of  finding  particular  so- 
lutions may  be  extended  to  other  differential  equations. 

232.    Examples. 
1.  Find  the  particular  solution  of  the  equation 

t  +  xD^x  —  d,x  ^  {x^-\-t^  —  a^).         (13  f ) 

Solution.     This  equation,  freed  from  radicals,  becomes 

whence  (12  f)  becomes 

X  {t  +  xp)  =ip  (x^  +  i^  —  «-)• 

The  elimination  of  /;  gives  for  equation 

(x^  —  a^)  (x2  4-^2_a2)  =  0, 

of  which  the  factor 

x2  _!_  ^2  _  ^2  —  0  (14  f) 

is  the  particular  solution. 
25 


290  INTEGRAL    CALCULUS.  [b.  V.    CH.   XlfL 

Particular  solutions. 

2.    Find  the  particular  solution  of  the  equation 

x  —  tD,x-\-P,  (15  f) 

in  which  P  is  a  given  function  of  Z>,  x. 

Ans.    It  is   the  equation  obtained  by  the  elimination  of  2? 
between  the  equation 

^^  <    (I6f) 

and  t  -\-  DpP'  —  0,  ^    ^        ' 

in  which  P'  is  the  value  of  P  obtained  by  the  substitution  of 
p  for  jD^  X. 


THE   END. 


/•/</..;  .»■'»/; 


JAMES    MUNROE    AND    COMPANY'S    PUBLICA-TIONi. 

PEIRCE'S  COURSE 


OF 


PURE    MATHEMATICS. 

A  Course  of  Instruction  in  Pure  Mathematics,  for  the  Use 
of  Students.  By  Benjamin  Peirce,  A.  M.,  Perkins 
Professor  of  Mathematics  and  Astronomy  in  Harvard 
University. 

1.  An  Elementary  Treatise  on  Plane  and  Solid 
Geometry.     1  vol.  12mo.,  with  plates.     2d  edition. 

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as  we  can  see,  leaves  notliing  to  be  desired  in  this  branch  of  mathematics. 
The  doctrine  of  parallel  lines  as  presented  by  Mr.  Peirce  is  concise,  intel- 
ligible, and  in  our  judgment  entirely  satisiactory.  But  this  is  not  the 
only  improvement.  Every  page  shows  the  same  power  of  condensing,  and 
the  same  neatness  and  elegance,  for  which  the  two  works  on  Trigonometry, 
by  the  same  author,  are  so  remarkable.' — North  American  Review. 

2.  An  Elementary  Treatise  on  Algebra.    To  which 

are   added   Exponential   Equations    and    Logarithms. 

3d  edition.     12mo. 

The  editor  of  the  Christian  Examiner,  after  speaking  of  the  Algebra, 
concludes  by  saying :  '  We  can  say  nothing  better  for  the  book  than  that 
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ject, will  fijid  many  new  things  in  this  treatise  deserving  their  attention; 
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ical Trigonometry,  with  their  Applications  to  Navi- 
gation, Surveying,  Heights  and  Distances,  and  Spheri- 
cal Astronomy,  and  particularly  adapted  to  explaining 
the  Construction  of  Bowditch's  Navigator,  and  the 
Nautical  Almanac.     3d  edition,  12mo,  with  plates. 

'The  work,  of  which  we  give  the  title  above,  (Plane  and  Spherical  Trig- 
onometry,) is  part  of  a  course  of  elementary  mathematics,  which  he  (Prof. 
P.)  has  given  notice  that  he  intends  to  publish.  They  show,  throughout,  the 
marks  of  an  original  thinker.  But  in  this  work  there  is  a  unity  and  homoge- 
neousness,  which  shows  that  it  is  not  mere  compilation,  but  that  it  has 
passed  through  and  been  reproduced  by  the  author's  own  mind.  The 
analysis  is  conducted  throughout  in  the  most  finished  and  elegant  manner. 
Both  these  works  are  remarkable  for  brevity  and  simplicity  (qualities 
which  instructers  will  know  how  to  prize) ;  and  we  believe  they  will  be 
found  fully  equal,  if  not  superior,  to  any  works  now  in  use,  for  the  purpose 
for  which  they  were  designed.' — North  American  Review. 

*  As  a  text-book  for  such  a  course  of  instruction  as  is  usually  taught  in 
our  Colleges,  it  (the  Plane  Trigonometry)  seems  to  be  superior  to  any  be- 
fore published  on  that  subject;  and  if  the  projected  course  of  elementary 
treatises  be  carried  out  in  the  same  spirit  and  style,  there  is  no  doubt  they 
•will  be  highly  useful  to  both  teachers  and  pupils.' — Mathematical  Miscti- 
lany. 

[X^  This  work  is  used  in  the  Naval  Schools,  as  a  text-book,  and  in  many 
of  oar  Universitieg. 


JAMES    MUNROE    AND    COMPANY'S    PUBLICATION!. 

PEIRCE'S  COUESB 


OF 


PURE    MATHEMATICS. 

A  Course  of  Instruction  in  Pure  Mathematics,  for  the  Use 
of  Students.  By  Benjamin  Peirce,  A.  M.,  Perkins 
Professor  of  Mathematics  and  Astronomy  in  Harvard 
University. 

1.  An  Elementary  Treatise  on  Plane  and  Solid 
Geometry.     1  vol.  12mo.,  with  plates.     2d  edition. 

*  The  book  is  througliOTit  simple,  though  neat  and  concise ;  and,  as  far 
as  -vve  can  see,  leaves  nothing  to  be  desired  in  this  branch  of  mathematics. 
The  doctrine  of  parallel  lines  as  presented  bv  Mr.  Peirce  is  concise,  intel- 
ligible, and  in  ovu*  judgment  entirely  satisfactory.  But  this  is  not  the 
only  improvement.  Every  page  shows  the  same  power  of  condensing,  and 
the  same  neatness  andelegance,  for  which  the  two  works  on  Trigonometry, 
by  the  same  author,  are  so  remarkable.' — North  American  Review. 

2.  An  Elementary  Treatise  on  Algebra.    To  which 

are   added   Exponential   Equations    and    Logarithms. 

3d  edition.     12mo. 

The  editor  of  the  Christian  Examiner,  after  speaking  of  the  Algebra, 
concludes  by  saying :  '  We  can  say  nothing  better  for  the  book  than  that 
he  (Prof  P.)  prepared  it,  and  that  it  fully  sustains  the  reputation  for 
science  which  he  has  already  won.  Those  who  are  interested  in  the  sub- 
ject, will  find  many  new  thmgs  in  this  treatise  deserving  their  attention; 
particularly  the  polynomial  theorem  of  Arbogast.' 

3.  An  Elementary  Treatise  on  Plane  and  Spher- 
ical Trigonometry,  with  their  Applications  to  Navi- 
gation, Surveying,  Heights  and  Distances,  and  Spheri- 
cal Astronomy,  and  particularly  adapted  to  explaining 
the  Construction  of  Bowditch's  Navigator,  and  the 
Nautical  Almanac.     3d  edition,  12mo,  with  plates. 

•The  work,  of  which  we  give  the  title  above,  (Plane  and  Spherical  Trig- 
onometry,) is  part  of  a  course  of  elementary  mathematics,  which  he  (Prof. 
P.)  has  given  notice  that  he  intends  to  publish.  They  show,  throughout,  the 
marks  of  an  original  thinker.  But  in  this  work  there  is  a  unity  and  homoge- 
neousness,  which  shows  that  it  is  not  mere  compilation,  but  that  it  has 
passed  through  and  been  reproduced  by  the  author's  own  mind.  The 
analysis  is  conducted  throughout  in  the  most  finished  and  elegant  manner. 
Both  these  works  are  remarkable  for  brevity  and  simplicity  (qualities 
which  instructers  will  know  how  to  prize) ;  and  we  believe  they  will  be 
found  fully  equal,  if  not  superior,  to  any  works  now  in  use,  for  the  purpose 
for  which  they  were  designed.' — North  Ajnericaii  Review. 

*  As  a  text-book  for  such  a  course  of  instruction  as  is  usually  taught  in 
our  Colleges,  it  (the  Plane  Trigonometry)  seems  to  be  superior  to  any  be- 
fore published  on  that  subject;  and  if  the  projected  course  of  elementary 
treatises  be  carried  out  in  the  same  spirit  and  stvle,  there  is  no  doubt  they 
will  be  highly  useful  to  both  teachers  and  pupils.' — MatheTnatical  Miscel- 
lany. 

Ct^*  This  work  is  used  in  the  Naval  Schools,  as  a  text-book,  and  in  many 
of  oar  Universitieg. 


JAMES    MUNROE    AND    COMPANY'S    PUBLICATIONS. 

PEIRCE^S  COURSE 

OF 

NATURAL  PHILOSOPHY. 

Designed  for  the  Use  of  High  Schools  and  Colleges 
Compiled  by  Benjamin  Peirce,  A.  M.,  Perkins  Profes- 
sor of  Mathematics  and  Astronomy  in  Harvard  Uni- 
versity.    To  be  comprised  in  five  vols.  8vo. 

Vohimc  II  of  this  Course  is  now  published,  and  has  been  adopted  as  a 
text-book  in  the  University  at  Cambridge ;  and  contains 

An  Elementary  Treatise  on  Sound. 

'  Our  limits  do  not  allow  us  to  give  a  minute  review  of  this  excellent  work, 
which  is  distinguished  no  less  by  the  exactness  and  method  of  its  science 
than  the  simplicity  and  perspicuity  of  its  language.  Those,  who  can  com- 
prehend any  scientific  investigation  of  this  kind,  cannot  fail  to  tmderstand 
the  views  I'lere  given  of  a  subject  which  is  certainly  embarrassed  with 
many  difficulties,  and,  in  certain  particulars,  (to  use  the  language  applied 
by  Prof.  Peirce  to  one  portion  of  the  science,)  "  altogether  intractable." 

'  At  the  beginning  of  the  work  is  a  very  comprehensive  list  of  writers 
upon  Soimd  in  general,  as  well  as  musical  and  other  soxmds,  from  the  age 
of  Aristotle  to  the  present  day,  which  has  been  prepared  with  vast  labor 
and  industry,  and  is,  we  believe,  the  most  complete  catalogue  of  the  kind 
extant  in  any  language.  This  labor  alone  is  of  incalculable  value  to  those 
persons  who  are  desirous  of  pursuing  the  subject,  to  whom  we  take  great 
pleasvire  in  recommendmg  a  work  so  simple  and  inteUigiblCj  and,  at  the 
same  time,  so  thoroughly  scientific' — Scientific  and  Literary  Jounud. 

'  It  is  seldom  that  a  book  comes  from  the  press  which  is  designed  to 
meet  a  more  urgent  want  of  the  community  than  this  second  volume  of  a 
Course  of  Natural  Philosophy.  At  a  time  when  so  mnny  books,  good  and 
bad,  are  wi-itten,  on  eveiy  variety  of  subjects,  and  with  particular  adapta- 
tion to  the  widely  different  classes  of  readers  —  and  especially  when  the 
overflowing  supply  of  manuals  used  seems  to  leave  nothing  to  be  wanted 
in  the  work  of  instniction  —  it  is  a  little  singular  that  there  is  occasion  for 
the  remark  that  this  volume  fills  a  gap  which  no  one  before  appears  to 
have  noticed,  or,  at  any  rate,  to  have  endeavoi-ed  to  close.  In  elementary 
ti-eatises  prepared  exclusively  for  the  use  of  common  schools,  acoustics 
have  been  considered,  in  a  simple  manner,  among  the  other  branches  of 
Natural  Philosophy.  But  no  work  Avhatever  has  appeared  designed  for 
the  hisjher  places  of  insti-uction,  and  presenting  a  full  and  accurate  analy- 
sis of  the  principles  of  sound.  There  is  some  occasion,  then,  for  congi-atu- 
lation  that  we  have  a  i*eally  new  book,  and  one  which  cannot  be  laid  aside ; 
and  since  it  is  probably  destined  to  be  introduced  into  all  our  colleges,  as 
it  has  already  been  into  one,  we  are  glad  to  know  that  it  has  been  executed 
in  such  a  manner  as  Avill  leave  little  demand  for  another. 

*  Professor  Peirce  lays  no  claim  to  originality  in  this  work.  He  tells  us 
tliat  he  made  Sir  John  Herschell's  Treatise  on  Sound,  -vAi-itten  for  the  Ency 
clopaedia  Metropolitana,  the  basis  of  his  o^\ti  book.  Li  remodelling  that 
work,  he  has  consulted  all  the  works  on  Sound  of  any  consequence,  as  well 
as  embodied  the  very  important  discoveries  recently  made  by  Faraday ; 
in  a  word,  he  has  wrought  a  pleasing  and  symmetrical  whole  out  of  all  the 
loose  and  scattered  materials  which  relate  to  the  subject.  The  labor  of 
such  a  task  is  immense,  and  it  is  no  small  praise  to  say  that  it  has  been 
done  accurately,  and  leaves  nothing  more  to  be  desired. 

'  There  is  one  subject  connected  with  acoustics  which  is  extremely  diffi- 
cult, and  in  which  Ave  think  Professor  Peirce  has  been  remarkably  suc- 
cessful ;  tlie  organs  of  the  human  voice.  There  have  been  very  contradic- 
tory theories  in  regard  to  the  peculiar  service  of  each  part  of  this  complex 
structure.  In  Mr.  Peirce's  book  it  is  shown  how  they  might  be  reconciled. 
^—Nortli  American  Review. 

3 


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